Spectral Multiplicity Entails Sample-wise Multiple Descent

1 INTRODUCTION The double/multiple descent phenomenon attracted recent research attention due to ( Belkin et al. , 2019 ) . This line of work focuses on the parameter-wise double/multiple descent phenomenon of the risk curve ( Bartlett et al. , 2020 ; Tsigler & Bartlett , 2020 ; Belkin et al. , 2019 ; 2020 ; Chen et al. , 2020a ; Liang et al. , 2020 ; Advani et al. , 2020 ; Bös & Opper , 1998 ; Krogh & Hertz , 1992 ; Le Cun et al. , 1991 ; Mei & Montanari , 2019 ; Opper et al. , 1990 ; Vallet et al. , 1989 ; Watkin et al. , 1993 ) . The classical learning theory shows that when the number of parameters ( which reflects the model complexity ) increases , the test error ( generalization risk ) first decreases due to more fitting power , and then increases due to overfitting . The generalization risk attains a peak at the interpolation threshold ( the number of parameters equals the number of data points so that the model interpolates the data ) . This results in a U-shaped risk curve if we plot the test error versus the number of parameters . The double descent risk curve posits that the risk will decrease ( again ) if one further increases the model complexity beyond the interpolation threshold ( Belkin et al. , 2019 ) . Thus there is a second descent in addition to the first one in the U-shaped stage of the curve . Belkin et al . ( 2019 ) presented empirical results and showed the existence of such double descent behavior in the random Fourier features model , the fully connected neural network , and the random forest model . Prior to ( Belkin et al. , 2019 ) , earlier studies of the shape and features of the risk curve in a number of contexts include ( Vallet et al. , 1989 ; Opper et al. , 1990 ; Le Cun et al. , 1991 ; Krogh & Hertz , 1992 ; Bös & Opper , 1998 ; Watkin et al. , 1993 ; Advani et al. , 2020 ) . Loog et al . ( 2020 ) presented a prehistory of the double descent phenomenon . Belkin et al . ( 2020 ) proved the double descent curve in the Gaussian model and the Fourier series model . Mei & Montanari ( 2019 ) theoretically established the double descent curve of the random features regression . Bartlett et al . ( 2020 ) ; Tsigler & Bartlett ( 2020 ) characterized the conditions for ridgeless and ridge linear regression problems , respectively , under which the minimum-norm interpolants achieve near-optimal generalization risk . Liang et al . ( 2020 ) showed that the test error of the minimum-norm interpolator of data in reproducing kernel Hilbert space is upper bounded by a multiple descent curve as the model complexity increases . They also presented a numerical result supporting that the test error itself exhibits a multiple descent curve . Chen et al . ( 2020a ) proved that the multiple descent curve does exist for the minimum-norm interpolator in linear regression and that the curve can be even designed . Following the parameter-wise double descent , research interest extended to epoch-wise and samplewise double descent ( Nakkiran et al. , 2020 ; Chen et al. , 2020b ; Min et al. , 2021 ; Nakkiran et al. , 2021 ) . Nakkiran et al . ( 2020 ) observed from their numerical result that the generalization risk experiences a double descent as one keeps the model size fixed and increases the training time . They called this observation epoch-wise double descent . Nakkiran et al . ( 2020 ) also noted sample-wise non-monotonicity , which means that more data can hurt generalization . Nakkiran et al . ( 2021 ) proved that for isotropic features , optimally regularized ridge regression yields a monotonic generalization risk curve with more samples . Nakkiran et al . ( 2021 ) also showed that if the features are formed by projecting high-dimensional isotropic data to a random low-dimensional space ( say , ddimensional ) , the optimally regularized ridge regression has a monotonic generalization risk curve with increasing d ( the model size ) . Sample-wise non-monotonicity and double descent was also observed in ( Chen et al. , 2020b ; Min et al. , 2021 ) in adversarially trained models . C ompared to ( Wu & Xu , 2020 ; ichi Amari et al. , 2021 ; Dobriban & Wager , 2018 ; Richards et al. , 2021 ) , in what follows , we highlight our contributions and the differences from them . First , our major contribution is providing a rigorous proof for the existence of sample-wise ( test error vs. the number of training samples ) double and multiple descent in linear regression . However , ( Richards et al. , 2021 ) only mentioned parameter-wise double descent ( test error vs. model capacity ) in their related work . ( ichi Amari et al. , 2021 ) only mentioned epoch-wise ( test error vs. training time ) double descent in Appendix A.2 . Neither ( Richards et al. , 2021 ) nor ( ichi Amari et al. , 2021 ) mentioned multiple descent . Second , we made and theoretically proved the observation that an ill-conditioned covariance matrix is a sufficient condition for the existence of sample-wise multiple descent . To the best of our knowledge , our work is the first paper that pointed this out . Third , we solved the Stieltjes transform explicitly and derived explicit formulae for the risk and variance in our setup . In addition , we also provided rigorous treatment to the ridgeless setting and also obtained explicit formulae for it . Fourth , there is another difference between our paper and the papers that the reviewer mentioned . ( Wu & Xu , 2020 ; ichi Amari et al. , 2021 ; Dobriban & Wager , 2018 ; Richards et al. , 2021 ) assumed a prior on the true linear model and takes expectation over the prior . In our paper , we do not assume a prior on the true linear model and our risk does not take the expectation over a random true linear model . In the setting of generally anisotropic features , this paper gives an asymptotic characterization of the generalization risk curve with more samples . The asymptotic regime is an approximation for large n , d and can also shed light on practical machine learning problems . We first introduce our problem setup . 1.1 PROBLEM SETUP Data Distribution Let Σ ∈ Rd×d be a positive semi-definite matrix which is termed the covariance matrix , and let θ∗ ∈ Rd . The eigenvalues of Σ are λ1 , . . . , λm with multiplicity d1 , . . . , dm , respectively . We have d = ∑m i=1 di . Assume that λ1 , . . . , λm are fixed , distinct , all positive , and do not depend on d ( i.e. , for all d , the eigenvalue of Σ are always λ1 , . . . , λm ) . We assume the following data distribution D for ( x , y ) ∈ Rd × R : x ∼ N ( 0 , Σ ) , y = x > θ∗ + , where x and are independent and ∼ N ( 0 , σ2 ) . In practice , there are natural random variables x that satisfy our assumption . For example , assume that we want to use machine A to measure the length of several objects and use machine B to measure their temperature . The measured lengths and temperatures follow an i.i.d . Gaussian distribution . However , the variance of measurement of machine A is different from that of machine B . Then we consider the random vector formed by the measurements x = ( l1 , . . . , ln , t1 , . . . , tn ) , where li and ti are the length and temperature of object i , respectively . This results in a block-structured covariance matrix . When we measure more objects , the size of the covariance matrix tends to infinity . Second , the motivation came from ( Nakkiran et al. , 2021 ) . ( Nakkiran et al. , 2021 ) observed empirically in their Figure 2 that when the covariance matrix has a block structure ( specifically , there are only two fixed different eigenvalues 10 and 1 ) , the expected excess risk exhibits multiple descent . We quantitatively studied this observation and obtained the related formulae . The excess risk of an estimator θ ∈ Rd is given by R ( θ ) = Ex , y∼D [ ( y − x > θ ) 2 − ( y − x > θ∗ ) 2 ] . Assume that the training data { ( xi , yi ) } ni=1 ⊆ Rd × R is drawn i.i.d . from D. Write X = x > 1 ... x > n ∈ Rn×d , y = y1 ... yn ∈ Rn . ( 1 ) We have y = Xθ∗ + , where ∼ N ( 0 , σ2In ) . Ridge Estimator and Minimum-Norm Estimator Definition 1 ( Ridge estimator ) . The ridge estimator θ̂λ , n , d ∈ Rd ( λ > 0 ) solves the following minimization problem min θ∈Rd 1 n ‖Xθ − y‖22 + λ ‖θ‖ 2 2 . Definition 2 ( Minimum-norm estimator ) . The minimum-norm estimator ( also known as the ridgeless estimator ) θ̂0 , n , d ∈ Rd solves the following minimization problem min θ∈Rd ‖θ‖2 such that ‖Xθ − y‖2 = min θ∈Rd ‖Xθ − y‖2 . We are interested in the expected excess risk of θ̂λ , n , d , which is given by Rλ , n , d = E [ R ( θ̂λ , n , d ) ] . The expectation is taken over the randomness of the training data { ( xi , yi ) } ni=1 . Asymptotic Regime Let Πi ∈ Rd×d be the orthogonal projection to the eigenspace of λi . This paper focuses on the asymptotic behavior of the expected excess risk of θ̂λ , n , d where n , di → +∞ , di/n→ zi ( zi is a fixed positive constant ) , and ‖Πiθ∗‖2 → ηi . In other words , we are interested in lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi Rλ , n , d . 1.2 OUR CONTRIBUTIONS Our contributions are summarized as follows . 1 . We obtain the formulae for the limiting bias and variance , and thereby the limiting risk . We use two methods to obtain these formulae . Specifically , we obtain the limiting bias and variance by solving the Stieltjes transform and computing its derivatives and antiderivatives . We also use convex Gaussian min-max theorem ( CGMT ) ( Thrampoulidis et al. , 2015 ) to compute the limiting variance . The advantage of the CGMT method is that it is more mathematically tractable for the ridgeless estimator . Through the CGMT approach , we obtain a closed-form formula for the variance in the underparameterized regime and simplify the formula for the variance in the overparameterized regime . Moreover , based on the simplified formula , we deduce a closed-form expression for the variance if the covariance matrix of the data distribution has two different eigenvalues . 2 . We find and theoretically prove that sample-wise multiple descent happens when the covariance matrix has eigenvalues of very different orders of magnitude ( thus the covariance matrix is highly ill-conditioned ) . 3 . We show that if the true linear model θ∗ satisfies ‖Πiθ∗‖2 = √ di d , optimal regulariza- tion ( i.e. , pick λ that minimizes the generalization risk of θ̂λ , n , d ) results in a monotone generalization risk curve—in other words , with optimal regularization , more data samples always improve generalization . Thus there is no sample-wise double or multiple descent . This provides a theoretical proof of a phenomenon observed in ( Nakkiran et al. , 2021 ) that optimal regularization can mitigate double descent for anisotropic data . Note that without regularization , there will be a blow-up in expected excess risk when n = d ( the linear model exactly interpolates the data ) and therefore , there is no samplewise descent across the under- and over-parameterized regimes . 2 PRELIMINARIES Notation Write [ m ] for { 1 , 2 , . . . , m } . Let i denote the imaginary unit . If x ∈ Rn and Σ ∈ Rn×n is a positive semidefinite matrix , write ‖x‖Σ , √ x > Σx . For a vector x , let ‖·‖1 and ‖·‖2 denote the ` 1 and ` 2 norm , respectively . Let denote the Hadamard ( entry-wise ) product between vectors . Write ‖ · ‖2 and ‖ · ‖F for the spectral matrix norm and Frobenius matrix norm , respectively . Let 4 denotes the Loewner order . For two square matrices A and B of the same size , write A 4 B if B − A is positive semidefinite . Define spec ( A ) as the set of all eigenvalues of A . Let O ( d ) = { A ∈ Rd×d | AA > = A > A = Id } denote the set of d × d orthogonal matrices . Define Sd−1 ( r ) , { x ∈ Rd | ‖x‖2 = r } . Denote almost sure convergence by a.s.→ , and convergence in probability plim and P→ . Ridge Estimator and Minimum-Norm Estimator We begin with the equivalent characterizations of the ridge and minimum-norm estimator . An equivalent characterization of the ridge estimator θ̂λ , n , d is θ̂λ , n , d = ( X > X + λnId ) −1 X > y = X > ( λnIn +XX > ) −1 y . ( 2 ) The second equality in Equation ( 2 ) is because of the Sherman–Morrison–Woodbury formula . A proof of Equation ( 2 ) can be found in ( Tsigler & Bartlett , 2020 ) . An equivalent definition of the minimum-norm estimator θ̂0 , n , d is that θ̂0 , n , d solves the following minimization problem min θ∈Rd ‖θ‖2 such that X > Xθ = X > y . Thus we have θ̂0 , n , d = ( X > X ) + X > y = X > ( XX > ) + y = X+y , where A+ denotes the pseudo-inverse of A . The second and third equalities are because of the identity X+ = ( X > X ) + X > = X > ( XX > ) + . The minimum-norm estimator is the limit of the ridge estimator θ̂λ , n , d as λ→ 0+ : θ̂0 , n , d = lim λ→0+ θ̂λ , n , d . This is because of the identity limλ→0+ ( X > X + λnId ) −1 X > = limλ→0+ X > ( λnIn +XX > ) −1 = X+ . Bias-Variance Decomposition of Expected Excess Risk We first show that the excess risk of an estimator θ equals the norm of θ − θ∗ : R ( θ ) = E ( x , y ) ∼D [ ( y − x > θ ) 2 − ( y − x > θ∗ ) ] = Ex [ ( x > ( θ∗ − θ ) ) 2 ] = E [ ( θ∗ − θ ) > Σ ( θ∗ − θ ) ] = E [ ‖θ∗ − θ‖2Σ ] . For the ridge estimator , the expected excess risk is Rλ , d , n =E [ ‖θ∗ −X > ( nλIn +XX > ) −1 ( Xθ∗ + ) ‖2Σ ] =E [ ‖ ( Id −X > ( nλIn +XX > ) −1X ) θ∗ −X > ( nλIn +XX > ) −1 ‖2Σ ] =E [ ‖ ( Id −X > ( nλIn +XX > ) −1X ) θ∗‖2Σ ] + E [ ∥∥X > ( nλIn +XX > ) −1 ∥∥2Σ ] =E [ ‖ ( Id −X > ( nλIn +XX > ) −1X ) θ∗‖2Σ ] + σ2E tr [ XΣX > ( nλIn +XX > ) −2 ] , Bλ , d , n + Vλ , d , n . ( 3 ) For the minimum-norm estimator , the expected excess risk is R0 , d , n = E [ ‖θ∗ −X+ ( Xθ∗ + ) ‖2Σ ] = E [ ∥∥ ( Id −X+X ) θ∗ −X+ ∥∥2Σ ] = E [ ∥∥ ( Id −X+X ) θ∗∥∥2Σ ] + E [ ∥∥X+ ∥∥2Σ ] = E [ ∥∥ ( Id −X+X ) θ∗∥∥2Σ ] + σ2E tr [ ( X+ ) > ΣX+ ] , B0 , d , n + V0 , d , n . ( 4 ) We call Bλ , d , n and B0 , d , n the bias term , and call Vλ , d , n and V0 , d , n the variance term . The bias and variance for the minimum-norm estimator are the limit of their counterpart for the ridge estimator as λ → 0+ , i.e. , limλ→0+ Bλ , d , n = B0 , d , n and limλ→0+ Vλ , d , n = V0 , d , n ( this can be shown by Lebesgue ’ s dominated convergence theorem , see our proof in Lemma 5 and Lemma 6 , respectively ) . 3 MAIN RESULTS 3.1 LIMITING RISK AND SAMPLE-WISE MULTIPLE DESCENT We study the limiting bias and variance for a linear regression problem in which the data distribution follows a multivariate normal distribution , the spectrum of the covariance matrix exhibits a block structure and tends to a discrete distribution . Thanks to the random matrix theory , we obtain the formulae ( presented in Theorem 1 ) for the limiting bias and variance , and thereby the total risk . We use two methods to obtain these formulae . The first method is through the Stieltjes transform of the matrix 1nXX > . The central quantity for computing the limiting bias and variance through the first method is the solution ρ∗ to the optimization problem Equation ( 5 ) in Item 1 of Theorem 1 . Item 1 guarantees the existence of a solution and determines its optimality condition Equation ( 6 ) . Item 2 computes the Jacobian matrix of ρ∗ with respect to λi and provides a closed-form formula to compute the Jacobian matrix . Equation ( 9 ) and Equation ( 10 ) in Item 4 give the formulae for the limiting bias obtained by the first method . Equation ( 11 ) and Equation ( 12 ) give the limiting variance . The second method is through the convex Gaussian min-max theorem ( CGMT ) ( Thrampoulidis et al. , 2015 ) . The central quantity is the solution r∗ to the minimax optimization problem Equation ( 8 ) in Item 3 . We use CGMT to obtain the formulae for the variance term . They are presented in Equation ( 13 ) and Equation ( 14 ) in Item 4 . Theorem 1 . The following statements hold : 1 . There exists a minimizer ρ ∈ Rm+ that solves inf ρ∈Rm+ log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) . ( 5 ) The minimizer ρ∗ satisfies λi λ+ ∑m j=1 λjρ ∗ j + 1− zi ρi = 0 , ∀i ∈ [ m ] . ( 6 ) 2 . Let ρ∗ ∈ Rm be a minimizer of Equation ( 5 ) and J = ∂ρ ∗ ∂λ ∈ R m×m be the Jacobian matrix Jij = ∂ρ∗i ∂λj . Then J is given by J = ( diag ( λ ) + ( λ+ λ > ρ∗ ) Im − ( z− ρ∗ ) λ > ) −1 ( ( z− ρ∗ ) ρ∗ > − diag ( ρ∗ ) ) and the matrix ( diag ( λ ) + ( λ+ λ > ρ∗ ) Im − ( z− ρ∗ ) λ > ) is always invertible . 3 . Define r = ( r1 , . . . , rm ) , λ = ( λ1 , . . . , λm ) , and ϑ ( rt , r , λ ) = 2rt √ 1 + ∑ i∈ [ m ] r2i − 2rt ∑ i∈ [ m ] √ ziri + ∑ i∈ [ m ] 1 λi r2i − λr2t . ( 7 ) For any Kt ≥ 2λ and Ku ≥ 2λ+ ( 2+ √ γ ) λ , we have max 0≤rt≤Kt min 0≤ri≤Ku ϑ ( rt , r , λ ) = min 0≤ri≤Ku max 0≤rt≤Kt ϑ ( rt , r , λ ) = max rt≥0 min ri≥0 ϑ ( rt , r , λ ) = min ri≥0 max rt≥0 ϑ ( rt , r , λ ) ( 8 ) and the above optimization problem has a solution . 4 . Let r∗ = ( r∗1 , . . . , r ∗ m ) solve Equation ( 8 ) . Define q = ( η21/z1 , . . . , η 2 m/zm ) > and view λ = ( λ1 , . . . , λm ) > as a column vector . The limiting bias is given by lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi Bλ , d , n = q > ( λ ρ∗ + Jλ 2 ) , ( 9 ) lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi B0 , d , n = lim λ→0+ q > ( λ ρ∗ + Jλ 2 ) . ( 10 ) The limiting variance is given by lim n , di→+∞ di/n→zi Vλ , d , n = σ 2λ 2 > ( ρ∗ + J > λ ) ( λ+ λ > ρ∗ ) 2 , ( 11 ) lim n , di→+∞ di/n→zi V0 , d , n = σ 2 lim λ→0+ λ 2 > ( ρ∗ + J > λ ) ( λ+ λ > ρ∗ ) 2 , ( 12 ) lim n , di→+∞ di/n→zi Vλ , d , n = σ 2 m∑ i=1 r∗2i , ( 13 ) lim n , di→+∞ di/n→zi V0 , d , n = σ 2 lim λ→0+ m∑ i=1 r∗2i . ( 14 ) Figure 1 illustrates the theoretical and numerical values of the bias , variance , and total risk . We observe a triple descent in Figure 1a where the covariance matrix has three blocks , and a quadruple descent in Figure 1b where the covariance has four blocks . In the three-block example , we set λ3 λ2 λ1 ( λ1 = 1 , λ2 = 100 , λ3 = 1000 ) . In the four-block example , we set λ4 λ3 λ2 λ1 ( λ1 = 1 , λ2 = 100 , λ3 = 104 , λ4 = 107 ) . For the values of other parameters , please refer to the caption of Figure 1 Our findings provide an explanation for the occurrence of sample-wise multiple descent : it occurs when the covariance matrix is highly ill-conditioned . Moreover , we find that the generalization risk curve is continuous in ridge regression ( λ > 0 ) while it blows up at n = d in ridgeless regression ( λ = 0 ) . We can see the singularity ( at n = d = 200 ) of the ridgeless generalization risk curve in Figure 2a . Following Theorem 1 , we focus on the variance in the ridgeless case ( λ = 0 ) and further study the expressions in Equation ( 13 ) and Equation ( 14 ) . We find that the variance exhibits sharply different behaviors in the underparameterized and overparameterized regimes . Recall that we will let n , di → +∞ and keep di/n → zi . Then d/n → ∑ i∈ [ m ] zi . If lim d/n = ∑ i∈ [ m ] zi > 1 , we are in the underparameterized regime . In this regime , the bias vanishes and therefore the risk equals the variance . If lim d/n < 1 , we are in the overparameterized regime . Theorem 2 . If d/n→ ∑ i∈ [ m ] zi > 1 and r ∗ = ( r∗1 , . . . , r ∗ m ) solves min ri≥0 ∑ i∈ [ m ] 1 λi r2i subject to √∑ i∈ [ m ] r2i + 1 = ∑ i∈ [ m ] √ ziri , then we have an optimality condition for r∗ : r∗i r∗j = λi λj · √ ziA ∗ − r∗i√ zjA∗ − r∗j , i , j ∈ [ m ] , ( 15 ) where A∗ = √∑ i∈ [ m ] r ∗2 i + 1 . Moreover , we have limn , di→+∞ di/n→zi V0 , d , n = σ 2 limλ→0+ ∑m i=1 r ∗2 i . If d/n→ ∑ i∈ [ m ] zi < 1 , then we have lim n , di→+∞ di/n→zi V0 , d , n = σ 2 ∑ i∈ [ m ] zi 1− ∑ i∈ [ m ] zi . . Corollary 1 . If m = 1 and d/n→ z1 > 1 , we have limn , di→+∞ di/n→zi V0 , d , n = σ 2 1 z1−1 . Proof . In the case m = 1 , we have r∗1 solves minr1≥0 1 λ1 r21 subject to √ r21 + 1 = √ z1r1 . The equality constraint gives r∗21 = 1 z1−1 . Then by Theorem 2 , the limiting variance is σ 2r∗21 = σ 2 1 z1−1 . In Theorem 2 , we find that in the underparameterized regime , r∗ solves an equality-constrained minimization problem . In the proof of Theorem 2 , we see that the equality constraint is feasible in the underparameterized regime but infeasible in the overparameterized regime . Moreover , we present an optimality condition for r∗ , which will be used in Theorem 3 to study the two-block ( m = 2 ) case . If the data distribution is isotropic ( which means that the covariance matrix is a scalar matrix ) , Collorary 1 shows that the limiting variance is σ2 1z1−1 , which agrees with ( Hastie et al. , 2019 , Theorem 1 ) . In the overparameterized regime , however , we find that the limiting variance does not depend on the spectrum { λ1 , . . . , λm } of the covariance matrix and only depends on the noise intensity σ and the ratios zi = lim di/n . This agrees with ( Hastie et al. , 2019 , Proposition 2 ) . In Theorem 3 , we study the case m = 2 and present a concrete closed-form formula for the limiting variance in the overparameterized regime . Recall that the limiting variance in the underparameterized regime has a closed-form σ2 ∑ i∈ [ m ] zi 1− ∑ i∈ [ m ] zi for general m , as shown in Theorem 2 . Theorem 3 . If m = 2 and d/n→ z1 + z2 > 1 , we have lim n , di→+∞ di/n→zi V0 , d , n = σ 2 q 2 + 1 q2 ( z1 − 1 ) + 2q √ z1z2 + z2 − 1 . where q = λ1 ( z1 − 1 ) + λ2 ( 1− z2 ) + √ ( λ1 ( z1 − 1 ) + λ2 ( 1− z2 ) ) 2 + 4λ1λ2z1z2 2λ2 √ z1z2 . ( 16 ) We illustrate the theoretical values predicted by Theorem 3 ( overparameterized regime ) and Theorem 2 ( underparameterized regime ) in Figure 2a and compare it to the numerical values . Corollary 2 ( Triple descent in the two-block case ) . Assume m = 2 , z1 = z2 , d/n→ ζ = 2z1 , and λ2/λ1 = % . Define f % ( ζ ) = limn , di→+∞ di/n→zi V0 , d , n . We have f ( ζ ) , lim % →+∞ fρ ( ζ ) = σ2 ζ1−ζ ζ < 1 , σ2 ( 1 ζ−1 + 2 2−ζ − 1 ) 1 < ζ < 2 σ2 2ζ−2 ζ > 2 . ( 17 ) There exists ζ1 , ζ2 , ζ3 , ζ4 and % 0 such that for all % > % 0 , we have f ′ % ( ζ1 ) < 0 , f ′ % ( ζ2 ) > 0 , f ′ % ( ζ3 ) < 0 , and f ′ % ( ζ4 ) < 0 . Proof . The case ζ < 1 is already given in Theorem 2 . In the sequel , assume ζ > 1 . Define q as in Equation ( 16 ) . We have q = ζ + √ ζ2 ( % + 1 ) 2 − 4ζ ( % − 1 ) 2 + 4 ( % − 1 ) 2 − ( ζ − 2 ) % − 2 2ζ % . Recall Theorem 3 , we get f % ( ζ ) = lim n , di→+∞ di/n→zi V0 , d , n = 2 ( q2 + 1 ) ζ ( q + 1 ) 2 − 2 ( q2 + 1 ) = 2σ2 ζ ( q+1 ) 2 q2+1 − 2 . Direct calculation yields lim % →+∞ f % ( ζ ) = σ2 ζ1−ζ ζ < 1 , σ2 ( 1 ζ−1 + 2 2−ζ − 1 ) 1 < ζ < 2 σ2 2ζ−2 ζ > 2 . , g ( ζ ) , lim % →+∞ f ′ % ( ζ ) = σ2 1 ( ζ−1 ) 2 ζ < 1 , σ2 ζ 2−2 ( ζ2−3ζ+2 ) 2 1 < ζ < 2 σ2 −2 ( ζ−2 ) 2 ζ > 2 . , The function g ( ζ ) > 0 if ζ ∈ ( √ 2 , 2 ) and we have g ( ζ ) < 0 if ζ < √ 2 or ζ > 2 . Pick ζ1 > 2 > ζ2 > √ 2 > ζ3 > 1 > ζ4 . Then we have g ( ζ1 ) < 0 , g ( ζ2 ) > 0 , g ( ζ3 ) < 0 , and g ( ζ4 ) > 0 . There exists % 0 such that for all % > % 0 , we have f ′ % ( ζ1 ) < 0 , f ′ % ( ζ2 ) > 0 , f ′ % ( ζ3 ) < 0 , and f ′ % ( ζ4 ) < 0 . Collorary 2 theoretically proves that there exists triple descent when m = 2 and λ2 λ1 . Note that a larger ζ = lim d/n reflects a relatively smaller n. If f ′ % ( ζ ) < 0 , then f % ( ζ ) decreases on a neighborhood of ζ and therefore the limiting variance increases with a relatively larger n. As n becomes relatively larger , we see an increasing stage , a decreasing stage , and finally an increasing stage in order in the overparameterized regime ( n < d ) . When we further increase n and enter the underparameterized regime , we observe a decreasing stage . We illustrate f ( ζ ) in Figure 2b . In Figure 2b , we observe two singularities at ζ = 1 and ζ = 2 . 3.2 OPTIMAL REGULARIZATION MONOTONIZES GENERALIZATION RISK CURVE Recall the definition of the ridge estimator in Definition 1 . Since this subsection concerns samplewise monotonicity , we add a subscript n toX and y ( they are defined by Equation ( 1 ) in Section 1.1 ) to emphasize that they consist of n data items . Therefore we write θ̂λ , n , d , arg min θ 1 n ‖yn −Xnθ‖22 + λ ‖θ‖ 2 2 . In this subsection , under an assumption , we show that optimal regularization ( i.e. , pick λ that minimizes the generalization risk of θ̂λ , n , d ) results in a monotone generalization risk curve—in other words , with optimal regularization , more data always reduces the generalization risk . The assump- tion is that ‖Πiθ∗‖2 = √ di d , i.e. , the squared norm of the projection of θ ∗ onto each eigenspace of the covariance matrix is proportional to the dimension of that eigenspace . ( Nakkiran et al. , 2021 ) showed by numerical results that optimal regularization can mitigate double descent for anisotropic data distribution . We give a partial theoretical proof of their observed phenomenon . To ease the notation , we use γi , limn/di rather than zi , lim di/n in Theorem 4 because a larger γ reflects a relatively larger n ( in the limit ) . Theorem 4 shows that with the optimal regularization , the limiting risk is an increasing function of γ1 , . . . , γm . Theorem 4 ( Optimal regularization ) . If ‖Πiθ∗‖2 = √ di d , ( 18 ) then there exists a function g ( γ1 , . . . , γm ) such that g ( γ1 , . . . , γm ) is increasing in every γi and lim n , di→∞ n/di→γi inf λ > 0 EXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2 Σ = g ( γ1 , . . . , γm ) . 4 CONCLUSION We studied the generalization risk ( test error ) versus the number of training samples in ridgeless regression . Under the assumption that the data distribution is Gaussian and the spectrum distribution of its covariance matrix converges to a discrete distribution , we obtained the exact formulae for the limiting bias and variance terms using the random matrix theory when the dimension and the number of training samples go to infinity in a proportional manner . 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Reviews of Modern Physics , 65 ( 2 ) :499 , 1993 . Denny Wu and Ji Xu . On the optimal weighted ` 2 regularization in overparameterized linear regression . arXiv preprint arXiv:2006.05800 , 2020 . A EIGENDECOMPOSITION AND MORE NOTATION Write Σ = PΛP > , where P is an orthogonal matrix and Λ = diag ( λ1Id1 , . . . , λmIdm ) ∈ Rd×d is a diagonal matrix . Write λ− = mini∈ [ m ] λi and λ+ = maxi∈ [ m ] λi . We can generate x1 , . . . , xn from standard normal random vector zi ∼ N ( 0 , Id ) by setting xi = PΛ1/2zi . Therefore , if Z = ( z1 . . . zn ) ∈ Rd×n , we get X > = ( x1 . . . xn ) = PΛ 1/2 ( z1 . . . zn ) = PΛ 1/2Z . Take the transpose gives X = Z > Λ1/2P > . Note that every entry of Z ∈ Rd×n follows i.i.d . N ( 0 , 1 ) . Write Z in a row-partitioned form Z = Z > 1 ... Z > m , where Zi ∈ Rn×di . Write P in a column-partitioned form P = ( P1 . . . Pm ) , where Pi ∈ Rd×di . Recall that Πi ∈ Rd×d denotes the orthogonal projection to the eigenspace of λi . We have Πi = PiP > i . Define θ ′ , P > θ∗ and write it in a row-partitioned form θ′ = P > 1 θ ∗ ... P > mθ ∗ = θ ′ 1 ... θ′m , ( 19 ) where θ′i ∈ Rdi . Then ‖θ′i‖2 = ∥∥P > i θ∗∥∥2 = ∥∥PiP > i θ∗∥∥2 = ‖Πiθ∗‖2 . We summarize part of the notation above in Table 1 . B BIAS AND VARIANCE UNDER EIGENDECOMPOSITION Lemma 1 characterizes the smallest and largest eigenvalue of Z > Z d ( if n/d → γ < 1 ) and ZZ > n ( if n/d→ γ > 1 ) . Recall that we study the asymptotic regime di/n→ zi . Therefore γ = 1∑ j∈ [ m ] zj . Lemma 1 ( ( Bai & Yin , 2008 , Theorem 2 ) ) . Let Z ∈ Rd×n be a random matrix whose entries are i.i.d . N ( 0 , 1 ) random variables . As n , d→ +∞ , n/d→ γ ∈ ( 0 , 1 ) , we have limλmin ( Z > Z d ) = ( 1−√γ ) 2 , limλmax ( Z > Z d ) = ( 1 + √ γ ) 2 almost surely . If γ ∈ ( 1 , ∞ ) , as n , d→ +∞ , n/d→ γ , we have limλmin ( ZZ > n ) = ( 1− √ 1/γ ) 2 , limλmax ( ZZ > n ) = ( 1 + √ 1/γ ) 2 almost surely . Lemma 2 ( Corollary 5.35 ( Vershynin , 2010 ) ) . Let A be an N × n matrix whose entries are independent standard normal random variables . Then for every t ≥ 0 , with probability at least 1− 2 exp ( −t2/2 ) one has √ N − √ n− t ≤ smin ( A ) ≤ smax ( A ) ≤ √ N + √ n+ t , where smin ( A ) and smax ( A ) are the smallest and largest singular value of A. Lemma 3 . Let Z ∈ Rd×n be a random matrix whose entries are i.i.d . N ( 0 , 1 ) random variables , where d = d ( n ) satisfies limn→∞ nd ( n ) = γ . There exists universal positive constants C1 , C2 , N such that for all n > N , we have 0 < C1 < 1 n s2min ( Z ) ≤ 1 n s2max ( Z ) < C2 . Proof . Since d ( n ) n , with loss of generality , we assume n/d → γ ∈ ( 0 , 1 ) . Take t = c1 √ n in Lemma 2 , where c1 = 12 ( 1√ γ − 1 ) > 0 . With probability at least 1− 2e−c21n/2 , we have √ d− √ n− c1 √ n ≤ smin ( Z ) ≤ smax ( Z ) ≤ √ d+ √ n+ c1 √ n . Therefore , we deduce ( √ d n − 1− c1 ) 2 ≤ 1 n s2min ( Z ) ≤ 1 n s2max ( Z ) ≤ ( √ d n + 1 + c1 ) 2 . Define C1 = 18 ( 1√ γ − 1 ) 2 > 0 and C2 = ( 3√ γ + 1 ) 2 . Then there exists a universal constant N1 such that for all n > N1 , with probability at least 1− 2e−c 2 1n/2 , we have 0 < C1 < 1 n s2min ( Z ) ≤ 1 n s2max ( Z ) < C2 . Define event En = { C1 < 1 ns 2 min ( Z ) ≤ 1ns 2 max ( Z ) < C2 } c . Then we have Pr { En } ≤ 2e−c 2 1n/2 . Since ∑ n≥1 Pr { En } ≤ ∑ n≥1 2e −c21n/2 < ∞ , then the probability that infinitely many of En occur is 0 , i.e. , Pr { lim sup n En } = 0 . Therefore , there exists a universal constant N2 such that for all n > N2 , En does not happen , in other words , 0 < C1 < 1 n s2min ( Z ) ≤ 1 n s2max ( Z ) < C2 holds . Lemma 4 . Let Z ∈ Rd×n be a random matrix whose entries are i.i.d . N ( 0 , 1 ) random variables , and let p be a fixed positive integer which is viewed as a constant and hidden in .. If n d , we have E tr ( ZZ > ) n2 , E tr ( ZZ > ) 2 n3 , and E ‖Z‖p2 . np/2 . Proof . We have E tr ( ZZ > ) = E ‖Z‖2F = ∑ i∈ [ d ] , j∈ [ n ] Ez2ij = nd n2 . Write Z = z > 1 ... z > d , where zi ∈ Rn and zi ∼ N ( 0 , In ) . We have E ( z > i zi ) 2 = E ‖zi‖42 = n ( n + 2 ) . For i 6= j , we deduce E ( z > i zj ) 2 = E ( ‖zi‖2 ‖zj‖2 u > v ) 2 where u , v ∼ Unif ( Sn−1 ) and ‖zi‖2 , ‖zj‖2 , u , v are independent . Then we get E ( ‖zi‖ ‖zj‖ s > i sj ) 2 = E ‖zi‖22 ‖zj‖ 2 2 ( s > i sj ) 2 = n2Eu21 = n2 · 1 n = n . As a result , we have E tr ( ZZ > ) 2 = E ∥∥ZZ > ∥∥2 F = ∑ i , j∈ [ d ] E ( z > i zj ) 2 = dn ( n+ 2 ) + ( d2 − d ) n n3 . By ( Vershynin , 2018 ) , there exists a universal constant C > 0 such that for any t > 0 , P { ‖Z‖2 > C ( √ n+ √ d+ t ) } < 2e−t 2 . Define K = C ( √ n+ √ d ) . Then we have P { ‖Z‖2 > K + t } < 2e −t2/C2 . ( 20 ) Recall Γ ( z ) = ∫∞ 0 xz−1e−xdx . Setting t = C √ u in the equation below yields∫ ∞ 0 e−t 2/C2tp−1dt . ∫ ∞ 0 e−uu p 2−1du = Γ ( p 2 ) 1 . Then we can bound the following integral∫ ∞ K P { ‖Z‖2 ≥ t } pt p−1dt = ∫ ∞ 0 P { ‖Z‖2 ≥ K + t } p ( t+K ) p−1 dt . ∫ ∞ 0 e−t 2/C2 ( t+K ) p−1 dt . ∫ ∞ 0 e−t 2/C2 ( tp−1 +Kp−1 ) dt = ∫ ∞ 0 e−t 2/C2tp−1dt+Kp−1 ∫ ∞ 0 e−t 2/C2dt .n p−1 2 , where the first inequality is because of Equation ( 20 ) . We are in a position to bound E ‖Z‖p2 : E ‖Z‖p2 = ∫ ∞ 0 P { ‖Z‖p2 ≥ u } du = ∫ ∞ 0 P { ‖Z‖2 ≥ t } pt p−1dt = ∫ K 0 P { ‖Z‖2 ≥ t } pt p−1dt+ ∫ ∞ K P { ‖Z‖2 ≥ t } pt p−1dt .np/2 + n ( p−1 ) /2 .np/2 , where the first inequality is because∫ K 0 P { ‖Z‖2 ≥ t } pt p−1dt ≤ ∫ K 0 ptp−1dt = Kp . np/2 . Lemma 5 . The following equation for the bias term Bλ , d , n ( defined in Equation ( 3 ) ) holds Bλ , d , n =E [ ‖Λ1/2 ( Id − Λ1/2Z ( nλIn + Z > ΛZ ) −1 Z > Λ1/2 ) θ′‖22 ] ( 21 ) =E [ ‖Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 θ′‖22 ] . ( 22 ) Moreover , we have |Bλ , d , n| . ‖θ∗‖22 and limλ→0+ Bλ , d , n = B0 , d , n . For all sufficiently large n and d such that n/d → γ ∈ ( 0 , 1 ) , we have 0 ≤ ddλBλ , d , n . ‖θ ∗‖22 . Therefore , { Bλ , d , n } is uniformly bounded and uniformly equicontinuous with respect to λ ∈ ( 0 , ∞ ) . Proof . Introduce the shorthand notation M = Λ1/2ZZ > Λ1/2 ∈ Rd×d , A = Id + 1nλM ∈ R d×d , N = nλIn + Z > ΛZ ∈ Rn×n , and Q = Id − Λ1/2ZN−1Z > Λ1/2 ∈ Rd×d . Because X > ( nλIn +XX > ) −1X = PΛ1/2Z ( nλIn + Z > ΛZ ) −1 Z > Λ1/2P > , we have Bλ , d , n =E [ ‖ ( Id − PΛ1/2Z ( nλIn + Z > ΛZ ) −1 Z > Λ1/2P > ) θ∗‖2PΛP > ] =E [ ‖Λ1/2 ( Id − Λ1/2Z ( nλIn + Z > ΛZ ) −1 Z > Λ1/2 ) θ′‖22 ] =E [ ‖Λ1/2Qθ′‖22 ] . Using the Sherman–Morrison-Woodbury formula yields N−1 = 1 nλ In − 1 ( nλ ) 2Z > Λ1/2 ( I + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 Λ1/2Z = 1 nλ ( In − Z > Λ1/2 ( nλId + Λ 1/2ZZ > Λ1/2 ) −1 Λ1/2Z ) = 1 nλ ( In − Z > Λ1/2 ( nλId +M ) −1 Λ1/2Z ) . ( 23 ) It follows that Q =Id − Λ1/2ZN−1Z > Λ1/2 =Id − 1 nλ Λ1/2Z ( In − Z > Λ1/2 ( nλId +M ) −1 Λ1/2Z ) Z > Λ1/2 =Id − M nλ ( Id − ( nλId +M ) −1M ) =Id − M nλ ( Id − ( nλId +M ) −1 ( nλId +M − nλId ) ) =Id −M ( nλId +M ) −1 = ( Id + 1 nλ M ) −1 =A−1 . Therefore , we deduce Bλ , d , n = E [ ‖Λ1/2 ( Id + 1 nλ M ) −1 θ′‖22 ] = E [ ‖Λ1/2A−1θ′‖22 ] . Because ∥∥Λ1/2∥∥ 2 . 1 and ∥∥∥ ( Id + 1nλΛ1/2ZZ > Λ1/2 ) −1∥∥∥ 2 ≤ 1 , we have ‖Λ1/2A−1θ′‖22 . ‖θ′‖ 2 2 = ‖θ ∗‖22 . Therefore |Bλ , d , n| . ‖θ∗‖22 . Moreover , by the dominated convergence theorem , lim λ→0+ Bλ , d , n = B0 , d , n . We compute the derivative of A−1 : dA−1 dλ = −A−1 dA dλ A−1 = MA−2 nλ2 . The matrix Mn = Λ1/2ZZ > Λ1/2 n ∈ R d×d is positive semidefinite and its d − n smallest eigenvalues are zeros . Its non-zero eigenvalues are the same as the non-zero eigenvalues of Z > ΛZ n . Because all eigenvalues of Z > ΛZ n are positive almost surely , the spectrum of M n consists of d− n zeros and the spectrum of Z > ΛZ n . We study the range of the spectrum of Z > ΛZ n . Because λ− Z > Z n 4 Z > ΛZ n 4 λ+ Z > Z n , we deduce λmin ( Z > ΛZ n ) ≥ λ−λmin ( Z > Z n ) → λ− ( 1− √ 1/γ ) 2 ( 24 ) λmax ( Z > ΛZ n ) ≤ λ+λmax ( Z > Z n ) → λ+ ( 1 + √ 1/γ ) 2 . ( 25 ) Define L1 = λ−λmin ( Z > Z n ) and L2 = λ+λmax ( Z > Z n ) . We get limn , d→+∞ n/d→γ < 1 L1 = λ− ( 1− √ 1/γ ) 2 , limn , d→+∞ n/d→γ < 1 L2 = λ+ ( 1 + √ 1/γ ) 2 and spec ( Z > ΛZ n ) ⊆ [ L1 , L2 ] . We bound ∥∥MA−3∥∥ 2 ∥∥MA−3∥∥ 2 =n ∥∥∥∥∥Mn ( Id + M nλ ) −3∥∥∥∥∥ 2 =n max s∈spec ( Mn ) s ( 1 + s/λ ) 3 =n max s∈ { 0 } ∪spec ( Z > ΛZ n . ) s ( 1 + s/λ ) 3 =n max s∈spec ( Z > ΛZ n . ) s ( 1 + s/λ ) 3 ≤n max s∈ [ L1 , L2 ] s ( 1 + s/λ ) 3 . We compute ddλ‖Λ 1/2A−1θ′‖22 : d dλ ‖Λ1/2A−1θ′‖22 = 1 nλ2 θ′ > ( A−1ΛMA−2 +MA−2ΛA−1 ) θ′ = 1 nλ2 ( A−1θ′ ) > ( ΛMA−1 +MA−1Λ ) ( A−1θ′ ) Next , we bound ∣∣ d dλ‖Λ 1/2A−1θ′‖22 ∣∣ : ∣∣∣∣ ddλBλ , d , n ∣∣∣∣ ≤ 1 nλ2 ∥∥MA−2ΛA−1 +A−1ΛMA−2∥∥ 2 ‖θ′‖22 ≤ 2 nλ2 ∥∥MA−2ΛA−1∥∥ 2 ‖θ′‖22 = 2 nλ2 ∥∥MA−3AΛA−1∥∥ 2 ‖θ′‖22 ≤ 2 nλ2 ∥∥MA−3∥∥ 2 ∥∥AΛA−1∥∥ 2 ‖θ′‖22 . 1 λ2 max s∈ [ L1 , L2 ] s ( 1 + s/λ ) 3 ‖θ ′‖22 , where the last inequality is because ∥∥AΛA−1∥∥ 2 = ‖Λ‖2 ≤ λ+ . 1 . Define f ( s ) = s ( 1+s/λ ) 3 . Because f ′ ( s ) = λ 3 ( λ−2s ) ( λ+s ) 4 , the function f is increasing on [ 0 , λ/2 ] and decreasing on [ λ/2 , +∞ ) . If λ ≤ 2L1 , we have max s∈ [ L1 , L2 ] s ( 1 + s/λ ) 3 = L1 ( 1 + L1/λ ) 3 . It follows that 1 λ2 · L1 ( 1 + L1/λ ) 3 = L1λ ( λ+ L1 ) 3 ≤ max λ∈ [ 0,2L1 ] L1λ ( λ+ L1 ) 3 . 1 L1 . If λ ≥ 2L2 , we get 1 λ2 max s∈ [ L1 , L2 ] s ( 1 + s/λ ) 3 = 1 λ2 · L2 ( 1 + L2/λ ) 3 ≤ max λ∈ [ 2L2 , ∞ ) L2λ ( λ+ L2 ) 3 . 1 L2 ≤ 1 L1 . If 2L1 < λ < 2L2 , we obtain 1 λ2 max s∈ [ L1 , L2 ] s ( 1 + s/λ ) 3 . 1 λ . 1 L1 . In all three cases , we show that 1λ2 maxs∈ [ L1 , L2 ] s ( 1+s/λ ) 3 . 1L1 . It follows that∣∣∣∣ ddλ‖Λ1/2A−1θ′‖22 ∣∣∣∣ . 1L1 ‖θ′‖22 = ‖θ ′‖22 λ−λmin ( Z > Z n ) ‖θ′‖22 λmin ( Z > Z n ) . By Lemma 3 , there exists a universal constant n0 such that for all n > n0 , one has 1 λmin ( Z > Z n ) . 1 . Thus we conclude that ∣∣∣∣ ddλ‖Λ1/2A−1θ′‖22 ∣∣∣∣ . ‖θ′‖22 . We can exchange differentiation and expectation and get d dλ Bλ , d , n = E [ d dλ ‖Λ1/2A−1θ′‖22 ] and ∣∣∣∣ ddλBλ , d , n ∣∣∣∣ = E [ ∣∣∣∣ ddλ‖Λ1/2A−1θ′‖22 ∣∣∣∣ ] . ‖θ′‖22 . Lemma 6 . The following equation for the variance term holds Vλ , d , n =σ 2E‖Λ1/2 ( λnId + Λ 1/2ZZ > Λ1/2 ) −1 Λ1/2Z‖22 =σ2E‖ΛZ ( λnIn + Z > ΛZ ) −1 ‖22 . Moreover , for all sufficiently large n and d such that n/d → γ 6= 1 , we have limλ→0+ Vλ , d , n = V0 , d , n , |Vλ , d , n| . 1 and ∣∣ d dλVλ , d , n ∣∣ . 1 . Therefore , { Vλ , d , n } is uniformly bounded and uniformly equicontinuous with respect to λ ∈ ( 0 , ∞ ) . Proof . As in the proof of Lemma 5 , defineM = Λ1/2ZZ > Λ1/2 ∈ Rd×d andN = nλIn+Z > ΛZ ∈ Rn×n . Recalling Σ = PΛP > and X = Z > Λ1/2P > , we have Vλ , d , n =σ 2E tr [ XΣX > ( nλIn +XX > ) −2 ] =σ2E tr [ Z > Λ2ZN−2 ] =σ2E tr [ N−1Z > Λ2ZN−1 ] =σ2E ∥∥ΛZN−1∥∥2 F . Recalling Equation ( 23 ) yields ΛZN−1 = 1 nλ ΛZ ( In − Z > Λ1/2 ( nλId +M ) −1 Λ1/2Z ) = 1 nλ Λ1/2 ( In −M ( nλId +M ) −1 ) Λ1/2Z =Λ1/2 ( nλId +M ) −1 Λ1/2Z . Define R = Λ1/2Z ∈ Rd×n . We get ( nλId +M ) −1 Λ1/2Z = ( nλId +RR > ) −1R . Notice that if 0 < a < b , then aId +RR > 4 bId +RR > . We deduce ( bId +RR > ) 2 − ( aId +RR > ) 2 = ( b2 − a2 ) Id + 2 ( b− a ) RR > < 0 . Thus ( bId +RR > ) 2 < ( aId +RR > ) 2 , which implies ( bId +RR > ) −2 4 ( aId +RR > ) −2 . We get R > ( bId +RR > ) −2R 4 R > ( aId +RR > ) −2R , tr ( R > ( bId +RR > ) −2R ) ≤ tr ( R > ( aId +RR > ) −2R ) Let λ0 ( · ) denote the smallest non-zero eigenvalue of a positive semidefinite matrix . We bound the Frobenius norm ∥∥∥ ( nλId +M ) −1 Λ1/2Z∥∥∥2 F = tr ( R > ( nλId +RR > ) −1R ) ≤ tr ( lim λ→0+ R > ( nλId +RR > ) −2R ) = tr ( R > R ) + = tr ( Z > ΛZ ) + . tr ( Z > Z ) + . It follows that∥∥ΛZN−1∥∥2 F = ∥∥∥Λ1/2 ( nλId +M ) −1 Λ1/2Z∥∥∥2 F . ∥∥∥ ( nλId +M ) −1 Λ1/2Z∥∥∥2 F . tr ( Z > Z ) + = tr ( ZZ > ) + . If n/d → γ < 1 , the matrix Z > Z is full-rank almost surely . Then , using the formula for the mean of inverse Wishart distribution , we have E tr ( Z > Z ) + = trE ( Z > Z ) −1 = tr ( In d−n−1 ) 1 . If n/d → γ > 1 , the matrix ZZ > is full-rank almost surely . Similarly , we have E tr ( ZZ > ) + = trE ( ZZ > ) −1 1 . By the dominated convergence theorem , we have limλ→0+ Vλ , d , n = V0 , d , n . Moreover , Vλ , d , n . E ∥∥ΛZN−1∥∥2 F . 1 . Next we bound ddλV ( θ̂ ) . Because dN−1 dλ = −N −1 dN dλN −1 = −nN−2 , we deduce d dλ ∥∥ΛZN−1∥∥2 2 = d dλ tr ( N−1Z > Λ2ZN−1 ) = −2n tr ( Z > Λ2ZN−3 ) ≤ 0 . On the other hand , we have tr ( Z > Λ2ZN−3 ) = tr ( N−3/2Z > Λ2ZN−3/2 ) . tr ( N−3/2Z > ΛZN−3/2 ) = tr ( Z > ΛZN−3 ) = ∑ s∈spec ( Z > ΛZ ) s ( λn+ s ) 3 . Because the number of non-zero eigenvalues of Z > ΛZ equals rank ( Z > ΛZ ) = n ∧ d n , we get∣∣∣∣ ddλ ∥∥ΛZN−1∥∥22 ∣∣∣∣ n tr ( Z > Λ2ZN−3 ) . n2 max s∈spec ( Z > ΛZ ) s ( λn+ s ) 3 = max s∈spec ( Z > ΛZ n ) \ { 0 } s ( λ+ s ) 3 . If γ < 1 , the matrix Z > ΛZ n is full-rank almost surely . By Equation ( 24 ) and Equation ( 25 ) in the proof of Lemma 5 , there exists universal positive constants C1 and C2 such that spec ( Z > ΛZ n ) ⊆ [ C1 , C2 ] for all sufficiently large n and d such that n/d→ γ < 1 . If γ > 1 , the non-zero eigenvalues of Z > ΛZ n and M n are the same . The matrix M n is full-rank almost surely . Thus spec ( Z > ΛZ n ) \ { 0 } = spec ( M n ) . Because y > Λ−1y . y > y , λmin ( M n ) = min x6=0 x > Λ 1/2ZZ > Λ1/2 n x x > x = min y 6=0 y > ZZ > n y y > Λ−1y & min y 6=0 y > ZZ > n y y > y = λmin ( ZZ > n ) . Similarly , we get λmax ( M n ) . λmax ( ZZ > n ) . Therefore , there exists universal positive constants C1 and C2 such that spec ( Z > ΛZ n ) \ { 0 } ⊆ [ C1λmin ( Z > Z n ) , C2λmax ( Z > Z n ) ] . Thus in both cases , we have shown that there exists universal positive constants C1 and C2 such that spec ( Z > ΛZ n ) \ { 0 } ⊆ [ C1λmin ( Z > Z n ) , C2λmax ( Z > Z n ) ] . Define L1 = C1λmin ( Z > Z n ) and L2 = C2λmax ( Z > Z n ) . As a result , we get∣∣∣∣ ddλ ∥∥ΛZN−1∥∥22 ∣∣∣∣ . maxs∈ [ L1 , L2 ] s ( λ+ s ) 3 . Define f ( s ) = s ( λ+s ) 3 . Because f ′ ( s ) = λ−2s ( λ+s ) 4 , the function f is increasing on [ 0 , λ/2 ] and decreasing on [ λ/2 , +∞ ) . If λ ≥ 2L2 or λ ≤ 2L1 , we get max s∈ [ L1 , L2 ] s ( λ+ s ) 3 ≤ L1 ( λ+ L1 ) 3 ∨ L2 ( λ+ L2 ) 3 ≤ 1 L21 . If 2L1 < λ < 2L2 , we get max s∈ [ L1 , L2 ] s ( λ+ s ) 3 . 1 λ2 . 1 L21 . As a result , for all sufficiently large n , we have∣∣∣∣ ddλ ∥∥ΛZN−1∥∥22 ∣∣∣∣ = maxs∈ [ L1 , L2 ] s ( λ+ s ) 3 . 1L21 . 1λ2min ( Z > Zn ) . 1 , where the final inequality is because of Lemma 3 . We can exchange the expectation and differentiation and obtain d dλ Vλ , d , n = σ 2E d dλ ∥∥ΛZN−1∥∥2 2 and ∣∣∣∣ ddλVλ , d , n ∣∣∣∣ ≤ σ2E ∣∣∣∣ ddλ ∥∥ΛZN−1∥∥22 ∣∣∣∣ . 1 . C LEMMAS ON STIELTJES TRANSFORM Definition 3 ( Stieltjes transform ) . The Stieltjes transform of a distribution with cumulative distribution function F is defined by sF ( z ) = ∫ 1 λ− z dF ( λ ) ( z ∈ H , { z ∈ C | =z > 0 } ) . Lemma 7 ( Theorem 4.3 ( Bai & Silverstein , 2010 ) ) . Suppose that the entries of Xn ∈ Cn×p are complex random variables that are independent for each n and identically distributed for all n and satisfy E [ |x11 − Ex11|2 ] = 1 . Also , assume that Tn = diag ( τ1 , . . . , τp ) , τi is real , and the empirical distribution function of { τ1 , . . . , τp } converges almost surely to a probability distribution function H as n → ∞ . The entries of both Xn and Tn may depend on n , which is suppressed for brevity . Set Bn = An + 1nXnTnX ∗ n , where X ∗ n is the conjugate transpose of Xn , An is Hermitian , n× n satisfying FAn → FA almost surely , where FA is a distribution function ( possibly defective ) on the real line . Assume also that Xn , Tn , and An are independent . When p = p ( n ) with p/n → y > 0 as n → ∞ , then , almost surely , FBn , the empirical spectral distribution of the eigenvalues of Bn , converges vaguely , as n → ∞ , to a ( nonrandom ) distribution function F , where for any z ∈ C+ = { z ∈ C | =z > 0 } , its Stieltjes transform s = s ( z ) is the unique solution in C+ to the equation s = sA ( z − y ∫ τdH ( τ ) 1 + τs ) , where sA is the Stieltjes transform of FA . Lemma 8 . If the functions fα , gα : I → R satisfy fα ( x ) − gα ( x ) → 0 uniformly as α→ +∞ , then limα→+∞ ( infx∈I f ( x ) − infx∈I g ( x ) ) = 0 . Proof . Because fα ( x ) − gα ( x ) → 0 uniformly as α→ +∞ , we have for ∀ > 0 , there exists N ( ) such that for ∀α > N ( ) and ∀x ∈ I , it holds that |fα ( x ) − gα ( x ) | < . Therefore , we get gα ( x ) − < fα ( x ) < gα ( x ) + . Thus we obtain inf x∈I fα ( x ) ≤ fα ( x ) < gα ( x ) + inf x∈I gα ( x ) − ≤ gα ( x ) − < fα ( x ) , which in turn implies inf x∈I fα ( x ) ≤ inf x∈I gα ( x ) + inf x∈I gα ( x ) − ≤ inf x∈I fα ( x ) . It follows that |infx∈I fα ( x ) − infx∈I gα ( x ) | ≤ . In other words , we proved lim α→+∞ ( inf x∈I f ( x ) − inf x∈I g ( x ) ) = 0 . Lemma 9 . Define N = λnIn + Z > ΛZ . Then we have lim n , di→+∞ di/n→zi tr ( N−1 ) = d dλ inf ρ∈Rm+ log λ+ ∑ i∈ [ m ] λiρi + ∑ i∈ [ m ] ( ρi − zi ( log ρi zi + 1 ) ) , ( 26 ) lim n , di→+∞ di/n→zi 1 n log det N n = inf ρ∈Rm+ log λ+ ∑ i∈ [ m ] λiρi + ∑ i∈ [ m ] ( ρi − zi ( log ρi zi + 1 ) ) . ( 27 ) Proof . Proof of Equation ( 26 ) . We apply Lemma 7 with An = 0n×n , Xn = Z > ∈ Rn×d , Tn = Λ , and Bn = 1nZ > ΛZ . The distribution function of 0n×n converges to 1t≤0 and its Stieltjes transform is sA ( z ) = ∫ 1 λ−zd1λ≤0 = − 1 z . The empirical distribution function of { λ1 , . . . , λ1︸ ︷︷ ︸ d1 , . . . , λm . . . , λm︸ ︷︷ ︸ dm } is Hn , di ( t ) = ∑ i∈ [ m ] di d 1t≤λi . Recall di/n → zi . Thus di/d → zi/K , where d/n → y = ∑ j∈ [ m ] zj . The empirical distribution function converges to H ( t ) = ∑ i∈ [ m ] zi y 1t≤λi . Then the empirical spectral distribution of the eigenvalues of 1 nZ > ΛZ converges vaguely to a nonrandom distribution function F and its Stieltjes transform is s = s ( z ) = lim n , di→+∞ di/n=zi 1 n tr ( 1 n Z > ΛZ − zIn ) −1 = lim n , di→+∞ di/n=zi tr ( Z > ΛZ − znIn ) −1 ( this is because of ( Bai & Silverstein , 2010 , Theorem B.9 ) ) . By Lemma 7 , s ( z ) is the unique solution in C+ to the equation s ( z ) = sA ( z − y ∫ τdH ( τ ) 1 + τs ) = − 1 z − ∑ i∈ [ m ] λizi 1+λis ( z ) , which gives s ( z ) z − ∑ i∈ [ m ] λizi 1 + λis ( z ) = −1 . We want to prove Equation ( 26 ) first . The lefthand side of Equation ( 26 ) equals lim n , di→+∞ di/n=zi tr ( λnIn + Z > ΛZ ) −1 = s ( −λ ) . Because the matrix 1nZ > ΛZ is positive semidefinite and thereby all of its eigenvalues are nonnegative , its limiting spectral distribution is supported on [ 0 , ∞ ) . The Stieltjes transform s ( z ) of the limiting spectral distribution can be continuously extended to ( −∞ , 0 ) . Therefore , for ∀λ > 0 , s ( −λ ) is the unique solution to the following equation s ( −λ ) ( λ+ m∑ i=1 λizi 1 + λis ( −λ ) ) = 1 . ( 28 ) We will verify that d dλ inf ρ∈Rm+ log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) satisfies Equation ( 28 ) . Take a minimizer ρ∗ of Equation ( 5 ) . Using the envelope theorem yields d dλ inf ρ∈Rm+ log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) = 1 λ+ ∑m j=1 λjρ ∗ j . ( 29 ) Plugging the righthand side of Equation ( 29 ) into Equation ( 28 ) , we get 1 λ+ ∑m j=1 λjρ ∗ j ( λ+ m∑ i=1 λizi 1 + λi · 1λ+∑mj=1 λjρ∗j ) = 1 . Rewriting the above equation yields m∑ i=1 λizi 1 + λi · 1λ+∑mj=1 λjρ∗j = m∑ i=1 λiρ ∗ i . It suffices to show that each summand on the lefthand side equals its counterpart on the righthand side λizi 1 + λi · 1λ+∑mj=1 λjρ∗j = λiρ ∗ i . We need to show zi ρ∗i = 1 + λi · 1 λ+ ∑m j=1 λjρ ∗ j , which is equivalent to Equation ( 6 ) and therefore holds . Hence we have proved Equation ( 26 ) . Proof of Equation ( 27 ) . We use α to denote the indices n , di . Define h ( λ ) = inf ρ∈Rm+ log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) . First , we want to show that limλ0→+∞ ( h ( λ0 ) − log λ0 ) = 0 . Define lλ0 ( ρ ) = log 1 + 1 λ0 m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) , q ( ρ ) = m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) . The Hessian matrix of q ( p ) is diag ( z1 ρ21 , . . . , zmρ2m ) , which is positive definite since zi , ρi > 0 . Therefore , q ( p ) is convex and the minimum of q ( ρ ) on Rm+ is attained at ρ = z , where z = ( z1 , . . . , zm ) > . The minimum is infρ∈Rm+ q ( ρ ) = q ( z ) = 0 . Because lim‖ρ‖2→+∞ lλ0 ( ρ ) = +∞ , there exists a universal constant K1 > ‖z‖2 > 0 such that lλ0 ( ρ ) > lλ0 ( z ) for all ‖ρ‖2 > K1 . Define E = { ρ ∈ Rm+ | ‖ρ‖2 ≤ K1 } . We have z ∈ E , infρ∈E lλ0 ( ρ ) = infρ∈Rm+ lλ0 ( ρ ) , and infρ∈E q ( ρ ) = infρ∈Rm+ q ( ρ ) = 0 . Therefore , we get h ( λ0 ) − log λ0 = inf ρ∈Rm+ lλ0 ( ρ ) = inf ρ∈E lλ0 ( ρ ) − inf ρ∈E q ( ρ ) . ( 30 ) On E , there exists a universal constant K2 > 0 such that ∑ j∈ [ m ] λjρj < K2 . Thus on E , we deduce 0 < lλ0 ( ρ ) − q ( ρ ) = log 1 + 1 λ0 m∑ j=1 λjρj < log ( 1 + K2 λ0 ) . The right-hand side log ( 1 + K2λ0 ) → 0 as λ0 → +∞ . Thus limλ0→+∞ ( lλ0 ( ρ ) − q ( ρ ) ) = 0 uniformly for ρ ∈ E. By Lemma 8 , we get lim λ0→+∞ ( inf ρ∈E lλ0 ( ρ ) − inf ρ∈E q ( ρ ) ) = 0 . Recalling Equation ( 30 ) yields lim λ0→+∞ ( h ( λ0 ) − log λ0 ) = 0 . ( 31 ) Define fα ( λ ) = 1n log det N n . Second , we want to show limα fα ( λ ) = h ( λ ) , where limα means limn , di→+∞ di/n=zi . We have fα ( λ ) − fα ( λ0 ) = ∫ λ λ0 f ′α ( x ) dx for ∀λ , λ0 > 0 . It follows that |fα ( λ ) − h ( λ ) | ≤ |fα ( λ ) − h ( λ ) + h ( λ0 ) − fα ( λ0 ) + fα ( λ0 ) − log λ0 + log λ0 − h ( λ0 ) | ≤ |fα ( λ ) − h ( λ ) + h ( λ0 ) − fα ( λ0 ) |+ |fα ( λ0 ) − log λ0|+ |log λ0 − h ( λ0 ) | = ∣∣∣∣∣ ∫ λ λ0 f ′α ( x ) dx− ( h ( λ ) − h ( λ0 ) ) ∣∣∣∣∣+ |fα ( λ0 ) − log λ0|+ |log λ0 − h ( λ0 ) | . Taking lim supα on both sides gives lim sup α |fα ( λ ) − h ( λ ) | ≤ lim sup α ∣∣∣∣∣ ∫ λ λ0 f ′α ( x ) dx− ( h ( λ ) − h ( λ0 ) ) ∣∣∣∣∣+lim supα |fα ( λ0 ) − log λ0|+|log λ0 − h ( λ0 ) | . ( 32 ) Recall f ′α ( λ ) = trN −1 and limα f ′α ( λ ) = h ′ ( λ ) ( this is exactly Equation ( 26 ) ) . Because∣∣trN−1∣∣ = trN−1 ≤ 1λ and ∫ λλ0 1xdx < +∞ , by the dominated convergence theorem , we have lim α ∫ λ λ0 f ′α ( x ) dx = ∫ λ λ0 h′ ( x ) dx = h ( λ ) − h ( λ0 ) . It follows that lim sup α ∣∣∣∣∣ ∫ λ λ0 f ′α ( x ) dx− ( h ( λ ) − h ( λ0 ) ) ∣∣∣∣∣ = limα ∣∣∣∣∣ ∫ λ λ0 f ′α ( x ) dx− ( h ( λ ) − h ( λ0 ) ) ∣∣∣∣∣ = 0 . ( 33 ) Since fα ( λ0 ) −log λ0 = 1 n log det ( λ0In + 1 n Z > ΛZ ) − 1 n log det ( λ0In ) = 1 n log det ( In + 1 nλ0 Z > ΛZ ) and the matrix 1nλ0Z > ΛZ is positive semidefinite , we have fα ( λ0 ) − log λ0 ≥ 0 . We have fα ( λ0 ) − log λ0 = 1 n log det ( In + 1 nλ0 Z > ΛZ ) ≤ 1 n log det ( In + λ+ nλ0 Z > Z ) ≤ log ( 1 + λ+ λ0 λmax ( Z > Z n ) ) ≤λ+ λ0 λmax ( Z > Z n ) . Then taking lim supα , we get lim sup α |fα ( λ0 ) − log λ0| = lim sup α ( fα ( λ0 ) − log λ0 ) ≤ λ+ λ0 lim sup α λmax ( Z > Z n ) . 1 λ0 , ( 34 ) where the last inequality is because lim supα λmax ( Z > Z n ) = ( 1 + √ γ ∨ 1γ ) 2 1 by Lemma 1 . Using Equation ( 32 ) , Equation ( 33 ) and Equation ( 34 ) gives lim sup α |fα ( λ ) − h ( λ ) | . 1 λ0 + |log λ0 − h ( λ0 ) | . Then taking limλ0→+∞ and recalling Equation ( 31 ) yields lim α |fα ( λ ) − h ( λ ) | = lim sup α |fα ( λ ) − h ( λ ) | = 0 . Therefore , we conclude limα fα ( λ ) = h ( λ ) . Lemma 10 . Define N = λnIn+Z > ΛZ = λnIn+ ∑ i∈ [ m ] λiZiZ > i . The following equation holds lim n , di→+∞ di/n→zi E [ ∂ ∂λi 1 n log det N n ] = ∂ ∂λi inf ρ∈Rm+ log λ+ ∑ i∈ [ m ] λiρi + ∑ i∈ [ m ] ( ρi − zi ( log ρi zi + 1 ) ) , ( 35 ) lim n , di→+∞ di/n→zi E [ ∂2 ∂λj∂λi 1 n log det N n ] = ∂2 ∂λj∂λi inf ρ∈Rm+ log λ+ ∑ i∈ [ m ] λiρi + ∑ i∈ [ m ] ( ρi − zi ( log ρi zi + 1 ) ) . ( 36 ) Proof . Proof of Equation ( 35 ) . We use α to denote the indices n , di and use limα to denote limn , di→+∞ di/n=zi . Define fα ( λi ) = E [ 1 n log det N n ] , f ′α ( λi ) = ∂ ∂λi E [ 1 n log det N n ] , and h ( λi ) = inf ρ∈Rm+ log λ+ ∑ i∈ [ m ] λiρi + ∑ i∈ [ m ] ( ρi − zi ( log ρi zi + 1 ) ) . We have∣∣∣∣ 1n log det Nn ∣∣∣∣ ≤ 1n log det ( λIn + λ+ Z > Z n ) = log λ+ 1 n log det ( In + λ+ nλ Z > Z ) . By Lemma 3 , there exists a universal constant C > 0 such that for all sufficiently large n , 1 n log det ( In + λ+ nλ Z > Z ) ≤ log ( 1 + C λ ) . Therefore , we get ∣∣∣∣ 1n log det Nn ∣∣∣∣ ≤ log ( λ+ C ) . By the dominated convergence theorem and Lemma 9 ( specifically , Equation ( 27 ) ) , we obtain lim α fα ( λi ) = h ( λi ) . ( 37 ) Because ∣∣∣∣ ∂∂λi 1n log det Nn ∣∣∣∣ = 1n tr ( Z > i N−1Zi ) ≤ 1λn2 tr ( Z > i Zi ) and E [ 1 λn2 tr ( Z > i Zi ) ] < +∞ , we can interchange the differentiation and the expectation and get f ′α ( λi ) = ∂ ∂λi E [ 1 n log det N n ] = E [ ∂ ∂λi 1 n log det N n ] . ( 38 ) Thus we deduce∣∣∣∣ ∂∂λiE [ 1 n log det N n ] ∣∣∣∣ = ∣∣∣∣E [ ∂∂λi 1n log det Nn ] ∣∣∣∣ ≤ E ∣∣∣∣ ∂∂λi 1n log det Nn ∣∣∣∣ ≤ E [ 1λn2 tr ( Z > i Zi ) ] . By Lemma 4 , E tr ( Z > i Zi ) n2 and therefore E [ 1 λn2 tr ( Z > i Zi ) ] . 1λ . The function sequence { f ′α } is uniformly bounded . Then we want to show that { f ′α } is uniformly equicontinuous by showing that { f ′′α } is uniformly bounded . Because∣∣∣∣ ∂2∂λ2i 1n log det Nn ∣∣∣∣ = 1n tr ( Z > i N−1Zi ) 2 ≤ 1nλ2 tr ( Z > i Zi n ) 2 and E [ 1 nλ2 tr ( Z > i Zi n ) 2 ] < +∞ , we can interchange the differentiation and the expectation and get ∂2 ∂λ2i E [ 1 n log det N n ] = ∂ ∂λi E [ ∂ ∂λi 1 n log det N n ] = E [ ∂2 ∂λ2i 1 n log det N n ] . Therefore , we deduce∣∣∣∣ ∂2∂λ2i E [ 1 n log det N n ] ∣∣∣∣ ≤ E ∣∣∣∣ ∂2∂λ2i 1n log det Nn ∣∣∣∣ ≤ 1nλ2E tr ( Z > i Zi n ) 2 . Again , by Lemma 4 , tr ( Z > i Zi n ) 2 n. It follows that 1nλ2E tr ( Z > i Zi n ) 2 . 1λ2 . Therefore { f ′ α } is uniformly equicontinuous . We want to show limα f ′α ( λi ) = h ′ ( λi ) by contradiction . If it is not true , there exists > 0 and a subsequence { f ′αk } such that ∣∣f ′αk ( λi ) − h′ ( λi ) ∣∣ ≥ . Let E = [ a , b ] 3 λi ( b > a > 0 ) be a closed interval that contains λi . The subsequence { f ′αk } is uniformly bounded and uniformly equicontinuous . By the Arzela-Ascoli theorem , there exists a subsequence { f ′αkj } that converges uniformly on λi ∈ E. Recall limα fα ( λi ) = h ( λi ) ( Equation ( 37 ) ) . Thus limj fαkj ( λi ) = h ( λi ) . By ( Rudin , 1976 , Theorem 7.17 ) , for λi ∈ E , we have lim j f ′αkj ( λi ) = h ′ ( λi ) . This is a contradiction . Hence , we have shown that limα f ′α ( λi ) = h ′ ( λi ) , which is exactly Equation ( 35 ) ( recall f ′α ( λi ) = ∂ ∂λi E [ 1 n log det N n ] = E [ ∂ ∂λi 1 n log det N n ] in Equation ( 38 ) ) . Proof of Equation ( 36 ) . Define gα ( λj ) = ∂∂λiE [ 1 n log det N n ] = E [ ∂ ∂λi 1 n log det N n ] . Then g′α ( λj ) = ∂2 ∂λj∂λi E [ 1 n log det N n ] = ∂∂λjE [ ∂ ∂λi 1 n log det N n ] . We have ∣∣∣∣ ∂2∂λj∂λi 1n log det Nn ∣∣∣∣ = 1 n tr ( ZiZ > i N −1ZjZ > j N −1 ) = 1 n tr ( Z > i N −1ZjZ > j N −1Zi ) = 1 n ∥∥Z > j N−1Zi∥∥2F ≤ 1 n ‖Zj‖22 ‖Zi‖ 2 2 ∥∥N−1∥∥2 F ≤ 1 λ2n2 ‖Zj‖22 ‖Zi‖ 2 2 . where the last inequality is because ∥∥N−1∥∥2 F ≤ ∥∥ 1 λnIn ∥∥2 F = 1λ2n . If i 6= j , by Lemma 4 , we have 1 λ2n2 E ‖Zj‖22 ‖Zi‖ 2 2 = 1 λ2n2 E ‖Zj‖22 · E ‖Zi‖ 2 2 . 1 λ2 . If i = j , by Lemma 4 , we have 1 λ2n2 E ‖Zi‖42 . 1 λ2n2 · n2 = 1 λ2 . As a result , we get ∣∣∣∣ ∂2∂λj∂λi 1n log det Nn ∣∣∣∣ . 1n · n2 · 1λ2n = 1λ2 . Thus we can interexchange ∂∂λj and expectation , and get g ′ α ( λj ) = E [ ∂2 ∂λj∂λi 1 n log det N n ] . Be- cause |g′α ( λj ) | ≤ E ∣∣∣ ∂2∂λj∂λi 1n log det Nn ∣∣∣ . 1λ2 , the function sequence { g′α } is uniformly bounded for λj . Define L = Z > j N −1Zi and W = Z > j N −1Zj . We have∣∣∣∣∣ ∂3∂λ2j∂λi 1n log det Nn ∣∣∣∣∣ = 2 n tr ( L > WL ) . 1 λn2 tr ( L > Z > j ZjL ) = 1 λn2 tr ( Z > i N −1 ( ZjZ > j ) 2N−1Zi ) . = 1 λn2 ∥∥ZjZ > j N−1Zi∥∥2F ≤ 1 λn2 ∥∥N−1∥∥2 F ∥∥ZjZ > j ∥∥22 ‖Zi‖22 ≤ 1 λ3n3 ∥∥ZjZ > j ∥∥22 ‖Zi‖22 = 1 λ3n3 ‖Zj‖42 ‖Zi‖ 2 2 , where the first inequality is becauseW 4 1λnZ > j Zj and the third inequality is becauseN −1 4 1λnIn and then ∥∥N−1∥∥2 F ≤ ∥∥ 1 λnIn ∥∥2 F ≤ 1λ2n . By Lemma 4 , we have E ‖Zj‖ 4 2 . n 2 and E ‖Zi‖22 . n. If i 6= j , then Zj and Zi are independent , and we deduce 1 λ3n3 E ‖Zj‖42 ‖Zi‖ 2 2 . 1 λ3 . If i = j , we have 1 λ3n3 E ‖Zi‖42 ‖Zi‖ 2 2 = 1 λ3n3 E ‖Zi‖62 . 1 λ3 . As a result , we deduce E [ ∂3 ∂λ2j∂λi 1 n log det N n ] = ∂∂λjE [ ∂2 ∂λj∂λi 1 n log det N n ] = g′′α ( λj ) . Moreover , we have |g′′α ( λj ) | ≤ E ∣∣∣∣∣ ∂3∂λ2j∂λi 1n log det Nn ∣∣∣∣∣ . 1λ3 . Therefore { g′α } is uniformly equicontinuous . Define w ( λj ) = ∂ ∂λi inf ρ∈Rm+ log λ+ ∑ i∈ [ m ] λiρi + ∑ i∈ [ m ] ( ρi − zi ( log ρi zi + 1 ) ) . We want to show by contradiction that limα g′α ( λj ) = w ′ ( λj ) . Assume that it is not true . Then there exists > 0 and a subsequence { g′αk } such that ∣∣g′αk ( λj ) − w′ ( λj ) ∣∣ > . Since { g′αk } is uniformly bounded and uniformly equicontinuous , by the Arzela-Ascoli theorem , there is a subsequence { g′αkr } that converges uniformly on a closed intervalE containing λj . Equation ( 35 ) shows that limα gα ( λj ) = w ( λj ) . It follows that limr gαkr ( λj ) = w ( λj ) . By ( Rudin , 1976 , Theorem 7.17 ) , for λi ∈ E , we have lim r g′αkr ( λj ) = w ′ ( λj ) , which is a contradiction . Therefore , we have shown that limα g′α ( λj ) = w ′ ( λj ) , which is exactly Equation ( 36 ) . D PROOF OF THEOREM 1 D.1 PROOF OF ITEM 1 Define g ( ρ ) = log ( λ+ ∑m j=1 λjρj ) + ∑m j=1 ( ρj − zj ( log ρjzj + 1 ) ) . The function g ( ρ ) is continuously differentiable on Rm+ . The boundary of R+m is ∂Rm+ = { ρ ∈ Rm | ( ∀i ∈ [ m ] , ρi ≥ 0 ) ∧ ( ∃i ∈ [ m ] , ρi = 0 ) } . Because limRm+3ρ→ρ0∈∂Rm+ g ( ρ ) = limRm+3ρ→∞ g ( ρ ) = +∞ , there exists a minimizer ρ ∗ ∈ Rm+ of g ( ρ ) . Taking the derivative with respect to ρi gives ∂ ∂ρi log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) = λi λ+ ∑m j=1 λjρj + 1− zi ρi . Setting it to zero gives Equation ( 6 ) . D.2 PROOF OF ITEM 2 Recall Equation ( 6 ) λi λ+ ∑m j=1 λjρ ∗ j + 1− zi ρ∗i = 0 , ∀i ∈ [ m ] . Rewriting the above equation gives ( zi − ρ∗i ) ( λ+ m∑ k=1 λkρ ∗ k ) = λiρ ∗ i , ∀i ∈ [ m ] . Rewriting it in the linear algebraic form yields ( z− ρ∗ ) ( λ+ λ > ρ∗ ) = λ ρ∗ . Applying ∂∂λ to both sides and using the implicit function theorem , we get ( z− ρ∗ ) ( ρ∗ > + λ > J ) − J ( λ+ λ > ρ∗ ) = diag ( λ ) J + diag ( ρ∗ ) . Arranging the above equation yields ( diag ( λ ) + ( λ+ λ > ρ∗ ) Im − ( z− ρ∗ ) λ > ) J = ( z− ρ∗ ) ρ∗ > − diag ( ρ∗ ) . Define a = λ + λ > ρ∗ , A = diag ( λ ) + ( λ+ λ > ρ∗ ) Im = diag ( λ ) + aIm and B = diag ( λ ) + ( λ+ λ > ρ∗ ) Im − ( z− ρ∗ ) λ > = A− ( z− ρ∗ ) λ > . The matrix determinant lemma gives det ( B ) = ( 1− λ > A−1 ( z− ρ∗ ) ) det ( A ) . Recall Equation ( 6 ) again and we have λi + a = zia ρ∗i . We have a− ∑ i∈ [ m ] λiρ ∗ i ( 1− ρ∗i /zi ) = λ+ ∑ i∈ [ m ] λiρ ∗ i − ∑ i∈ [ m ] λiρ ∗ i ( 1− ρ∗i /zi ) =λ+ ∑ i∈ [ m ] λi ( ρ∗i ) 2 zi > 0 . It follows that ∑ i∈ [ m ] λiρ ∗ i ( 1− ρ∗i /zi ) a < 1 . Then we compute λ > A−1 ( z− ρ∗ ) : λ > A−1 ( z− ρ∗ ) = ∑ i∈ [ m ] λi ( zi − ρ∗i ) λi + a = ∑ i∈ [ m ] λi ( zi − ρ∗i ) zia ρ∗i = ∑ i∈ [ m ] λiρ ∗ i ( 1− ρ∗i /zi ) a < 1 . Thus we get 1− λ > A−1 ( z− ρ∗ ) > 0 . Therefore , detB 6= 0 and the matrix B is invertible . D.3 PROOF OF ITEM 3 Lemma 11 . Define N = λnIn + Z > ΛZ , γ = ∑ i∈ [ m ] zi , r = ( r1 , . . . , rm ) , λ = ( λ1 , . . . , λm ) , and ϑ ( rt , r , λ ) = 2rt √ 1 + ∑ i∈ [ m ] r2i − 2rt ∑ i∈ [ m ] √ ziri + ∑ i∈ [ m ] 1 λi r2i − λr2t . For any Kt ≥ 2λ and Ku ≥ 2λ+ ( 2+ √ γ ) λ , we have lim n , di→+∞ di/n→zi trN−1 = lim n , di→+∞ di/n→zi E trN−1 = max 0≤rt≤Kt min 0≤ri≤Ku ϑ ( rt , r , λ ) = min 0≤ri≤Ku max 0≤rt≤Kt ϑ ( rt , r , λ ) = max rt≥0 min ri≥0 ϑ ( rt , r , λ ) = min ri≥0 max rt≥0 ϑ ( rt , r , λ ) . ( 39 ) If r∗ is a solution to the optimization problem in Equation ( 39 ) , then 1 + m∑ j=1 r∗2j = m∑ j=1 r∗j √ zj + λr ∗ t 2 , ( 40 ) r∗t r∗i√ 1 + ∑m j=1 r ∗2 j = r∗t √ zi − r∗i λi . ( 41 ) Moreover , we have ∂ ∂λi max rt≥0 min ri≥0 ϑ ( rt , r , λ ) = − r∗2i λ2i . Proof . Let g ∼ N ( 0 , In ) be a multivariate standard normal random vector . We have trN−1 =Egg > N−1g =Eg sup t∈Rn ( 2g > t− t > Nt ) =Eg sup t∈Rn ( 2g > t− t > Z > ΛZt− nλ ‖t‖22 ) =Eg sup t∈Rn inf u∈Rd ( 2g > t− 2u > ΛZt+ u > Λu− nλ ‖t‖22 ) =− 2Eg inf t∈Rn sup u∈Rd ( u > ΛZt− g > t− 1 2 u > Λu+ 1 2 nλ ‖t‖22 ) =− 2Eg inf t∈Rn sup u∈Rd ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) We view inf t∈Rn sup u∈Rd ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) ( 42 ) as the primal optimization ( PO ) problem in the convex Gaussian min-max theorem ( CGMT ) ( Thrampoulidis et al. , 2015 ) . The KKT conditions for Equation ( 42 ) give Z > u− g + nλt = 0 , Zt− Λ−1u = 0 . Solving the above equations gives t = N−1g , u = ΛZN−1g . With probability at least 1− 4 exp ( −cn ) ( c > 0 is a universal constant ) , we have ‖g‖2 ≤ 2 √ n and ‖Z‖ ≤ √ d+ 2 √ n ≤ ( 2 + √ γ ) √ n. Therefore , we get ‖t‖2 ≤ ∥∥N−1∥∥ ‖g‖2 ≤ 1λn · 2√n = 2λ√n , ‖u‖2 ≤ λ+ ‖Z‖ ‖t‖2 ≤ λ+ ( 2 + √ γ ) √ n · 2 λ √ n = 2λ+ ( 2 + √ γ ) λ . Write u = u1 ... um , where ui ∈ Rdi . For all Kt ≥ 2λ , Ku ≥ 2λ+ ( 2+√γ ) λ , the optimal solutions t∗ and u∗ to Equation ( 42 ) satisfy √ n ‖t∗‖2 ≤ Kt and ‖ui‖2 ≤ Ku for all i ∈ [ m ] with probability at least 1 − 4 exp ( −cn ) . Define St = { t ∈ Rn | √ n ‖t‖2 ≤ Kt } and Su = { u ∈ Rd | ‖ui‖ ≤ Ku , ∀i ∈ [ m ] } . We use α to denote the indices n , di and use limα to denote limn , di→+∞ di/n=zi . Define event Eα = { inf t∈Rn sup u∈Rd ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) = inf t∈St sup u∈Su ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) } . Then with probability at least 1 − 4 exp ( −cn ) , we have t∗ ∈ St and u∗ ∈ Su . Therefore the event Eα occurs with probability at least 1− 4 exp ( −cn ) , which yields P { Ecα } ≤ 4 exp ( −cn ) . Since ∑ n≥1 4 exp ( −cn ) < +∞ , by Borel-Cantelli lemma , we have P { lim sup α Ecα } = P { ( lim inf α Eα ) c } = 0 . Then with probability 1 , all but finitely many Eα occur . Then almost surely there exists n0 such that for all n > n0 , Eα occurs . The auxiliary optimization ( AO ) problem is inf t∈St sup u∈Su ( ‖t‖2 g > 1 u+ ‖u‖2 g > 2 t− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) = inf 0≤rt≤Kt sup 0≤ri≤Ku − ∥∥∥g −√∑i∈ [ m ] r2i g2∥∥∥ 2√ n rt + rt ∑ i∈ [ m ] ‖g1 , i‖2√ n ri − 1 2 ∑ i∈ [ m ] 1 λi r2i + 1 2 λr2t = inf 0≤rt≤Kt sup 0≤ri≤Ku −√1 + ∑ i∈ [ m ] r2i ‖g3‖2√ n rt + rt ∑ i∈ [ m ] ‖g1 , i‖2√ n ri − 1 2 ∑ i∈ [ m ] 1 λi r2i + 1 2 λr2t , where g1 ∼ N ( 0 , Id ) , g2 ∼ N ( 0 , In ) , and g3 ∼ N ( 0 , In ) . Taking n , di → +∞ with di/n→ zi constant , the strong law of large numbers gives√ 1 + ∑ i∈ [ m ] r2i ‖g3‖2√ n a.s.→ √ 1 + ∑ i∈ [ m ] r2i , ‖g1 , j‖2√ n = √ dj n ‖g1 , j‖2√ dj a.s.→ √zj . Define Xα ( rt , r ) = − √ 1 + ∑ i∈ [ m ] r2i ‖g3‖2√ n rt + rt ∑ i∈ [ m ] ‖g1 , i‖2√ n ri − 1 2 ∑ i∈ [ m ] 1 λi r2i + 1 2 λr2t . It is a stochastic process on ( rt , r ) ∈ [ 0 , Kt ] × [ 0 , Ku ] m. We have lim α Xα ( rt , r ) = X ( rt , r ) : = −rt √ 1 + ∑ i∈ [ m ] r2i + rt ∑ i∈ [ m ] √ ziri − 1 2 ∑ i∈ [ m ] 1 λi r2i + 1 2 λr2t almost surely . Since √ 1 + x2 is convex and increasing and the function ‖r‖2 is convex , thus √ 1 + ‖r‖22 is convex in r and then − √ 1 + ∑ i∈ [ m ] r 2 i ‖g3‖2√ n rt = − √ 1 + ‖r‖22 ‖g3‖2√ n rt is concave in r. Because − 12 ∑ i∈ [ m ] 1 λi r2i is concave in r and rt ∑ i∈ [ m ] ‖g1 , i‖2√ n ri is linear in r , we deduce that Xα ( rt , r ) is concave in r. By ( Liese & Miescke , 2008 , Lemma 7.75 ) , supr∈ [ 0 , Ku ] m |Xα ( rt , r ) −X ( rt , r ) | → 0 almost surely . Then for ∀ > 0 , there exists n0 ( ) , d0 , i ( ) , δ0 , i ( ) such that for all n > n0 ( ) , di > d0 , i ( ) , |di/n− zi| < δ0 , i ( ) and for all r ∈ [ 0 , Ku ] m , we have X ( rt , r ) − < Xα ( rt , r ) < X ( rt , r ) + . Thus we obtain X ( rt , r ) − < Xα ( rt , r ) ≤ sup r∈ [ 0 , Ku ] m Xα ( rt , r ) Xα ( rt , r ) < X ( rt , r ) + ≤ sup r∈ [ 0 , Ku ] m X ( rt , r ) + , which in turn implies sup r∈ [ 0 , Ku ] m X ( rt , r ) − ≤ sup r∈ [ 0 , Ku ] m Xα ( rt , r ) sup r∈ [ 0 , Ku ] m Xα ( rt , r ) ≤ sup r∈ [ 0 , Ku ] m X ( rt , r ) + . It follows that ∣∣∣supr∈ [ 0 , Ku ] m Xα ( rt , r ) − supr∈ [ 0 , Ku ] m X ( rt , r ) ∣∣∣ ≤ . In other words , we showed |Yα ( rt ) − Y ( rt ) | → 0 almost surely , where Y ( rt ) : = supr∈ [ 0 , Ku ] m Xα ( rt , r ) and Y ( rt ) : = supr∈ [ 0 , Ku ] m X ( rt , r ) . Because Xα ( rt , r ) is convex in rt , then Y ( rt ) = supr∈ [ 0 , Ku ] m Xα ( rt , r ) is convex in rt . By ( Liese & Miescke , 2008 , Lemma 7.75 ) again , suprt∈ [ 0 , Kt ] |Yα ( rt ) − Y ( rt ) | → 0 almost surely . A similar argument shows that∣∣∣∣ infrt∈ [ 0 , Kt ] Yα ( rt ) − infrt∈ [ 0 , Kt ] Y ( rt ) ∣∣∣∣ = ∣∣∣∣∣ infrt∈ [ 0 , Kt ] supr∈ [ 0 , Ku ] mXα ( rt , r ) − infrt∈ [ 0 , Kt ] supr∈ [ 0 , Ku ] mX ( rt , r ) ∣∣∣∣∣→ 0 almost surely . Therefore , we obtain inf t∈St sup u∈Su ( ‖t‖2 g > 1 u+ ‖u‖2 g > 2 t− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) = inf 0≤rt≤Kt sup 0≤ri≤Ku −√1 + ∑ i∈ [ m ] r2i ‖g3‖2√ n rt + rt ∑ i∈ [ m ] ‖g1 , i‖2√ n ri − 1 2 ∑ i∈ [ m ] 1 λi r2i + 1 2 λr2t a.s.→ inf 0≤rt≤Kt sup 0≤ri≤Ku −rt√1 + ∑ i∈ [ m ] r2i + rt ∑ i∈ [ m ] √ ziri − 1 2 ∑ i∈ [ m ] 1 λi r2i + 1 2 λr2t ( 43 ) = : µ . Define event Aα = { ∣∣∣∣ inft∈Rn supu∈Rd ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) − µ ∣∣∣∣ > τ } , Bα = { ∣∣∣∣ inft∈St supu∈Su ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) − µ ∣∣∣∣ > τ } , Cα = { ∣∣∣∣ inft∈St supu∈Su ( ‖t‖2 g > 1 u+ ‖u‖2 g > 2 t− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) − µ ∣∣∣∣ > τ } . Recall Eα = { inf t∈Rn sup u∈Rd ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) = inf t∈St sup u∈Su ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) } . We have Aα ∩ Eα ⊆ Bα . Equation ( 43 ) gives limα P { Cα } = 0 for any τ > 0 because almost sure convergence implies convergence in probability . By the convex Gaussian min-max theorem ( Thrampoulidis et al. , 2015 ) , we have P { Bα } ≤ 2P { Cα } . It follows that P { Aα } ≤ P { Aα ∩ Eα } + P { Ecα } ≤ P { Bα } + P { Ecα } ≤ 2P { Cα } + P { Ecα } . Taking lim supα on both sides , because lim supα P { Bα } ≤ 2 lim supα P { Cα } = 0 , we get lim sup α P { Aα } ≤ lim sup α P { Ecα } ≤ P { lim sup α Ecα } = 0 , where the second inequality is because of the reverse Fatou ’ s lemma . Thus inf t∈Rn sup u∈Rd ( u > Zt− g > t− 1 2 u > Λ−1u+ 1 2 nλ ‖t‖22 ) P→ µ . Therefore , we deduce g > N−1g P→ −2 inf 0≤rt≤Kt sup 0≤ri≤Ku −rt√1 + ∑ i∈ [ m ] r2i + rt ∑ i∈ [ m ] √ ziri − 1 2 ∑ i∈ [ m ] 1 λi r2i + 1 2 λr2t = sup 0≤rt≤Kt inf 0≤ri≤Ku 2rt√1 + ∑ i∈ [ m ] r2i − 2rt ∑ i∈ [ m ] √ ziri + ∑ i∈ [ m ] 1 λi r2i − λr2t = sup 0≤rt≤Kt inf 0≤ri≤Ku ϑ ( rt , r , λ ) . ( 44 ) Because ∣∣g > N−1g∣∣ ≤ 1λn ‖g‖22 and E 1λn ‖g‖22 = 1λ < ∞ , by the dominated convergence theorem for convergence in probability ( Cohn , 2013 , Proposition 3.1.6 ) , we get lim α trN−1 = lim α Eg [ g > N−1g ] = max 0≤rt≤Kt min 0≤ri≤Ku ϑ ( rt , r , λ ) . ( 45 ) Note that 2rt √ 1 + ∑ i∈ [ m ] r 2 i is convex in r , −2rt ∑ i∈ [ m ] √ ziri is linear in r , and ∑ i∈ [ m ] 1 λi r2i is strongly convex in r. Thus ϑ is strongly convex in r. Note that 2rt √ 1 + ∑ i∈ [ m ] r 2 i − 2rt ∑ i∈ [ m ] √ ziri is linear in rt and that −λr2t is strongly concave in rt . Thus ϑ is strongly concave in rt . Then ϑ has a unique saddle point ( r∗t , r ∗ ) on [ 0 , Kt ] × [ 0 , Ku ] m that satisfies max rt∈ [ 0 , Kt ] min r∈ [ 0 , Ku ] m ϑ ( rt , r ) = min r∈ [ 0 , Ku ] m max rt∈ [ 0 , Kt ] ϑ ( rt , r ) = ϑ ( r ∗ t , r ∗ ) , ( 46 ) where the first equality is due to Sion ’ s minimax theorem . Since ∣∣trN−1∣∣ ≤ 1λ , using the dominated convergence theorem and combining Equation ( 45 ) and Equation ( 46 ) yields lim α E trN−1 = max 0≤rt≤Kt min 0≤ri≤Ku ϑ ( rt , r , λ ) = min 0≤ri≤Ku max 0≤rt≤Kt ϑ ( rt , r , λ ) . By the uniqueness of the limit , the right-hand side max0≤rt≤Kt min0≤ri≤Ku ϑ ( rt , r , λ ) and min0≤ri≤Ku max0≤rt≤Kt ϑ ( rt , r , λ ) do not depend on Kt and Ku as long as Kt ≥ 2λandKu ≥ 2λ+ ( 2+ √ γ ) λ . Thus we have lim α E trN−1 = max rt≥0 min ri≥0 ϑ ( rt , r , λ ) = min ri≥0 max rt≥0 ϑ ( rt , r , λ ) . If r∗t = 0 , then ϑ ( 0 , r ∗ ) = minr∈ [ 0 , Ku ] m ∑ i∈ [ m ] 1 λi r2i = 0 . Thus r ∗ must be zero . However , ϑ ( 1 2λ ,0 ) = 34λ > ϑ ( 0 , r ∗ ) . Therefore r∗t > 0 . We compute the partial derivative ∂ϑ ∂ri = 2rt ri√ 1 + ∑ i∈ [ m ] r 2 i − 2rt √ zi + 2 ri λi . If r∗i = 0 , we have ∂ϑ ∂ri ∣∣∣ ri=0 , rt=r∗t = −2r∗t √ zi < 0 . Therefore , one can increase r∗i and make maxrt∈ [ 0 , Kt ] minr∈ [ 0 , Ku ] m ϑ ( rt , r ) smaller , which results in a contradiction . Thus r∗i > 0 . Thus the minimax value is attained when rt , ri > 0 for all i ∈ [ m ] . To obtain the optimality condition , we compute the partial derivatives ∂ϑ ∂rt = 2 √ 1 + ∑ i∈ [ m ] r2i − 2 ∑ i∈ [ m ] ri √ zi − 2λrt , ∂ϑ ∂ri = 2rt ri√ 1 + ∑ i∈ [ m ] r 2 i − 2rt √ zi + 2 ri λi . Setting them to zero gives the optimality condition for r∗t , r ∗ 1 , . . . , r ∗ m and yields Equation ( 40 ) and Equation ( 41 ) . Using the envelope theorem , we get ∂ ∂λi max rt∈ [ 0 , Kt ] min r∈ [ 0 , Ku ] m ϑ ( rt , r , λ ) = ∂ϑ ( r∗t , r ∗ , λ1 , . . . , λm ) ∂λi =− r ∗2 i λ2i . Lemma 12 . Define N = λnIn + Z > ΛZ . The following equation holds lim n , di→+∞ di/n→zi E [ ∂ ∂λi tr ( N−1 ) ] = ∂2 ∂λi∂λ inf ρ∈Rm+ log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) = −r∗2i λ2i , where r∗ is a solution to suprt > 0 infr1 , ... , rm > 0 ϑ ( rt , r1 , . . . , rm , λ1 , . . . , λm ) and ϑ ( rt , r1 , . . . , rm , λ1 , . . . , λm ) = 2rt √ 1 + ∑ i∈ [ m ] r2i − 2rt ∑ i∈ [ m ] √ ziri + ∑ i∈ [ m ] 1 λi r2i − λr2t . Proof . Since ∣∣∣∣ ∂∂λi tr ( N−1 ) ∣∣∣∣ = tr ( Z > i N−2Zi ) ≤ 1 ( λn ) 2 tr ( Z > i Zi ) ( 47 ) and E 1 ( λn ) 2 tr ( Z > i Zi ) 1λ2 ( by Lemma 4 ) , using the dominated convergence theorem gives E [ ∂ ∂λi tr ( N−1 ) ] = ∂ ∂λi E tr ( N−1 ) . We use α to denote the indices n , di and use limα to denote limn , di→+∞ di/n=zi . Define fα ( λi ) = E tr ( N−1 ) , g ( λi ) = ∂∂λ infρ∈Rm+ [ log ( λ+ ∑ i∈ [ m ] λiρi ) + ∑ i∈ [ m ] ( ρi − zi ( log ρizi + 1 ) ) ] , and h ( λi ) = suprt > 0 infr1 , ... , rm > 0 ϑ ( rt , r1 , . . . , rm , λ1 , . . . , λm ) . Because ∣∣tr ( N−1 ) ∣∣ ≤ tr ( 1 λnIn ) ≤ 1λ and limα tr ( N−1 ) = g ( λi ) ( by Lemma 9 ) , we have Lemma 9 limα fα ( λi ) = limα E tr ( N−1 ) = g ( λi ) . Lemma 11 shows limα fα ( λi ) = h ( λi ) . Therefore limα fα ( λi ) = g ( λi ) = h ( λi ) . Because of Equation ( 47 ) , we have f ′α ( λi ) = ∂ ∂λi E tr ( N−1 ) and |f ′α ( λi ) | = ∣∣∣∣ ∂∂λiE tr ( N−1 ) ∣∣∣∣ = ∣∣∣∣E [ ∂∂λi tr ( N−1 ) ] ∣∣∣∣ ≤ E ∣∣∣∣ ∂∂λi tr ( N−1 ) ∣∣∣∣ . 1λ2 and therefore { f ′α } is uniformly bounded for λi . Because∣∣∣∣ ∂2∂λ2i tr ( N−1 ) ∣∣∣∣ =2 tr ( N−1ZiZ > i N −1ZiZ > i N −1 ) . 1 λn tr ( N−1 ( ZiZ > i ) 2 N−1 ) = 1 λn tr ( ZiZ > i N −2ZiZ > i ) ≤ 1 ( λn ) 3 tr ( ZiZ > i ) 2 , and E 1 ( λn ) 3 tr ( ZiZ > i ) 2 1λ3 ( by Lemma 4 ) , using the dominated convergence theorem yields E [ ∂2 ∂λ2i tr ( N−1 ) ] = ∂ ∂λi E [ ∂ ∂λi tr ( N−1 ) ] = ∂2 ∂λ2i E [ tr ( N−1 ) ] = f ′′α ( λi ) . Moreover , we have |f ′′α ( λi ) | ≤ E ∣∣∣∣ ∂2∂λ2i tr ( N−1 ) ∣∣∣∣ . 1λ3 . Thus { f ′α } is uniformly equicontinuous for λi . We want to show that limα f ′α ( λi ) = g′ ( λi ) by contradiction . Assume that it is not true . Then there exists > 0 and a subsequence { f ′αk } such that ∣∣f ′αk ( λi ) − g′ ( λi ) ∣∣ > . Since { f ′αk } is uniformly bounded and uniformly equicontinuous for λi ∈ E ( E is any closed finite interval containing λi ) , by the Arzela-Ascoli theorem , there exists a subsequence { f ′αkr } that converges uniformly on E. Since limα fαkr ( λi ) = g ( λi ) , by ( Rudin , 1976 , Thoerem 7.17 ) , we have lim r f ′αkr ( λi ) = g ′ ( λi ) , which yields a contradiction . Therefore , we have limα f ′α ( λi ) = g ′ ( λi ) . Recall g ( λi ) = h ( λi ) for any λi > 0 . Then by the final part of Lemma 11 , we have limα f ′α ( λi ) = g ′ ( λi ) = h ′ ( λi ) = − r ∗2 i λ2i . D.4 BIAS Lemma 13 . Suppose that U ∼ ⊕ i∈ [ m ] Unif ( O ( di ) ) and V are two independent d × d random matrices such that V d= UV U > , where d = ∑m i=1 di . Let θ ∈ Rd be a fixed vector . Write θ = θ1 ... θm , where θi ∈ Rdi . Let φ ∼ ⊕ i∈ [ m ] Unif ( Sdi−1 ( ‖θi‖2 ) ) be a random vector independent of V and let Λ = diag ( λ1Id1 , . . . , λmIdm ) ∈ Rd×d . Then we have E [ ∥∥∥Λ1/2V θ∥∥∥2 2 ] = E [ ∥∥∥Λ1/2V φ∥∥∥2 2 ] . Proof . Recall UΛU > = Λ and noticing U > θ d= φ , we get E [ ∥∥∥Λ1/2V θ∥∥∥2 2 ] =E [ ∥∥∥Λ1/2UV U > θ∥∥∥2 2 ] =E [ θ > UV > U > ΛUV U > θ ] =E [ θ > UV > ΛV U > θ ] =E [ ∥∥∥Λ1/2V U > θ∥∥∥2 2 ] =E [ ∥∥∥Λ1/2V φ∥∥∥2 2 ] . Lemma 14 . Define Θ̃ = diag ( ‖θ′1‖22/d1Id1 , . . . , ‖θ′m‖22/dmIdm ) and S = Λ1/2Z ( nλIn + Z > ΛZ ) −1 Z > Λ1/2 . Then we have Bλ , d , n = ‖θ∗‖2Σ − 2E tr ( ΛSΘ̃ ) + E tr ( SΛSΘ̃ ) . Proof . Recall Equation ( 22 ) in Lemma 5 Bλ , d , n = E [ ‖Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 θ′‖22 ] . Let U ∼ ⊕ i∈ [ m ] Unif ( O ( di ) ) be a random matrix independent of Z . Because UZ d = Z , we have Id + 1 nλ Λ1/2ZZ > Λ1/2 d = Id + 1 nλ Λ1/2UZZ > U > Λ1/2 = U ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) U > . Define θ̃ ∼ ⊕ i∈ [ m ] Unif ( Sdi−1 ( ‖θ′i‖2 ) ) . Lemma 13 gives Bλ , d , n =E [ ‖Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 θ̃‖22 ] =E [ ‖Λ1/2 ( Id − S ) θ̃‖22 ] =E [ ‖ ( Id − S ) θ̃‖2Λ ] =E ∥∥∥θ̃∥∥∥2 Λ − E [ θ̃ > ΛSθ̃ ] − E [ θ̃ > SΛθ̃ ] + E [ θ̃ > SΛSθ̃ ] . Notice that ∥∥∥θ̃∥∥∥2 Λ = ‖θ′‖2Λ and Θ̃ = E [ θ̃θ̃ > ] . Because Θ̃ commutes with Λ , we have tr ( SΛΘ̃ ) = tr ( SΘ̃Λ ) = tr ( ΛSΘ̃ ) . In light of these , we deduce Bλ , d , n = ‖θ′‖ 2 Λ − E tr ( ΛSΘ̃ ) − E tr ( SΛΘ̃ ) + E tr ( SΛSΘ̃ ) = ‖θ′‖2Λ − 2E tr ( ΛSΘ̃ ) + E tr ( SΛSΘ̃ ) . Lemma 14 expresses the bias Bλ , d , n as the sum of three terms . Computing ‖θ′‖2Λ Note that ‖θ′‖ 2 Λ = θ ′ > Λθ′ = ∑ i∈ [ m ] λi ‖θ′i‖ 2 2 . Therefore , lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi ‖θ′‖2Λ = q > ( λ z ) . Computing E tr ( ΛSΘ̃ ) Define N = λnIn + Z > ΛZ = λnIn + ∑ i∈ [ m ] λiZiZ > i . We have E tr ( ΛSΘ̃ ) =E tr ( Z > Λ1/2Θ̃Λ3/2Z ( nλIn + Z > ΛZ ) −1 ) =E tr ( Z > Λ2Θ̃ZN−1 ) = ∑ i∈ [ m ] λ2i ‖θ′i‖22 di E tr ( ZiZ > i N −1 ) = ∑ i∈ [ m ] λ2i ‖θ′i‖22 n di E [ ∂ ∂λi 1 n log det N n ] = ∑ i∈ [ m ] λ2i ‖θ′i‖22 n di ∂ ∂λi E [ 1 n log det N n ] , where the second inequality is because Θ̃ commutes with Λ3/2 and the final equality is because of Equation ( 38 ) . Taking lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi and using Lemma 10 gives lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi E tr ( ΛSΘ̃ ) = ∑ i∈ [ m ] λ2i η 2 i zi ∂ ∂λi inf ρ∈Rm+ log λ+ ∑ i∈ [ m ] λiρi + ∑ i∈ [ m ] ( ρi − zi ( log ρi zi + 1 ) ) . Using the envelope theorem yields ∂ ∂λi inf ρ∈Rm+ log λ+ ∑ i∈ [ m ] λiρi + ∑ i∈ [ m ] ( ρi − zi ( log ρi zi + 1 ) ) = ρ∗i λ+ ∑ i∈ [ m ] λiρ ∗ i = zi − ρ∗i λi , where the final equality is because of Equation ( 6 ) in Item 1 . Therefore , we deduce lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi E tr ( ΛSΘ̃ ) = ∑ i∈ [ m ] λ2i η 2 i zi zi − ρ∗i λi = ∑ i∈ [ m ] λiη 2 i ( 1− ρ ∗ i zi ) = q > ( λ ( z− ρ∗ ) ) . Computing E tr ( SΛSΘ̃ ) We have E tr ( SΛSΘ̃ ) =E tr [ Λ1/2ZN−1Z > Λ2ZN−1Z > Λ1/2Θ̃ ] =E tr [ Z > Λ1/2Θ̃Λ1/2ZN−1Z > Λ2ZN−1 ] =E tr [ Z > ΛΘ̃ZN−1Z > Λ2ZN−1 ] = ∑ i∈ [ m ] λi‖θ′i‖22 di ∑ j∈ [ m ] λ2jE tr [ ZiZ > i N −1ZjZ > j N −1 ] =− ∑ i∈ [ m ] λi‖θ′i‖22n di ∑ j∈ [ m ] λ2jE [ ∂2 ∂λj∂λi 1 n log det N n ] , where the third equality is because Θ̃ commutes with Λ1/2 . Taking lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi and using Lemma 10 gives lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi E tr ( SΛSΘ̃ ) = − ∑ i∈ [ m ] λiη 2 i zi ∑ j∈ [ m ] λ2j ∂2 ∂λj∂λi inf ρ∈Rm+ log λ+ ∑ l∈ [ m ] λlρl + ∑ l∈ [ m ] ( ρl − zl ( log ρl zl + 1 ) ) . Write λ = ( λ1 , . . . , λm ) > and z = ( z1 , . . . , zm ) > . Let ρ∗ ∈ Rm be a minimizer of Equation ( 5 ) and J = ∂ρ ∗ ∂λ ∈ R m×m be the Jacobian matrix Jij = ∂ρ∗i ∂λj . Recall Item 2 ( diag ( λ ) + ( λ+ λ > ρ∗ ) Im − ( z− ρ∗ ) λ > ) J = ( z− ρ∗ ) ρ∗ > − diag ( ρ∗ ) . Using the envelope theorem , we have ∂ ∂λi inf ρ∈Rm+ log λ+ ∑ l∈ [ m ] λlρl + ∑ l∈ [ m ] ( ρl − zl ( log ρl zl + 1 ) ) = ρ∗i λ+ ∑ l∈ [ m ] λlρ ∗ l = ρ∗i λ+ λ > ρ∗ . Recall Equation ( 6 ) yields ρ∗i λ+ λ > ρ∗ = zi − ρ∗i λi . Differentiating the above equation with respect to λj gives ∂2 ∂λj∂λi inf ρ∈Rm+ log λ+ ∑ l∈ [ m ] λlρl + ∑ l∈ [ m ] ( ρl − zl ( log ρl zl + 1 ) ) = ∂ ∂λj zi − ρ∗i λi = −λiJij − ( zi − ρ∗i ) δij λ2i . It follows that lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi E tr ( SΛSΘ̃ ) = ∑ i∈ [ m ] λiη 2 i zi ∑ j∈ [ m ] λ2j λiJij + ( zi − ρ∗i ) δij λ2i = ∑ i , j∈ [ m ] qiλ 2 j ( Jij + ( zi − ρ∗i ) δij λi ) =q > ( λ ( z− ρ∗ ) + Jλ 2 ) Putting all three terms together , we have lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi Bλ , d , n = q > ( λ z ) −2q > ( λ ( z− ρ∗ ) ) +q > ( λ ( z− ρ∗ ) + Jλ 2 ) = q > ( λ ρ∗ + Jλ 2 ) . Since { Bλ , d , n } is uniformly bounded and uniformly equicontinuous for λ ∈ ( 0 , 1 ] by Lemma 5 , { Bλ , d , n } can be extended continuously to [ 0 , 1 ] and the family of extended functions is still uniformly bounded and uniformly equicontinuous for λ ∈ [ 0 , 1 ] . By the Arzela-Ascoli theorem , { Bλ , d , n } converges uniformly to the limit . By the Moore-Osgood theorem , we can exchange the two limits lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi and limλ→0+ and get lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi B0 , d , n = lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi lim λ→0+ Bλ , d , n = lim λ→0+ lim n , di→+∞ di/n→zi ‖Πiθ∗‖2→ηi Bλ , d , n = q > ( λ ρ∗ + Jλ 2 ) |λ=0 . D.5 VARIANCE Define N = nλIn + Z > ΛZ . Recalling Lemma 6 gives Vλ , d , n =σ2E‖ΛZN−1‖22 =σ2 m∑ i=1 λ2iE tr ( ZiZ > i N −2 ) =− σ2 m∑ i=1 λ2iE [ ∂ ∂λi tr ( N−1 ) ] . Using Lemma 12 , we get lim n , di→∞ di/n→zi Vλ , d , n =− σ2 m∑ i=1 λ2i lim n , di→∞ di/n→zi E [ ∂ ∂λi tr ( N−1 ) ] =− σ2 m∑ i=1 λ2i lim n , di→∞ di/n→zi E [ ∂ ∂λi tr ( N−1 ) ] =− σ2 m∑ i=1 λ2i ∂2 ∂λi∂λ inf ρ∈Rm+ log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) . Using the envelope theorem , we deduce ∂ ∂λ inf ρ∈Rm+ log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) = 1 λ+ ∑m j=1 λjρ ∗ j . Then we take ∂∂λi and obtain ∂2 ∂λi∂λ inf ρ∈Rm+ log λ+ m∑ j=1 λjρj + m∑ j=1 ( ρj − zj ( log ρj zj + 1 ) ) = ∂ ∂λi 1 λ+ ∑m j=1 λjρ ∗ j =− ρ∗i + ∑ j∈ [ m ] λjJji ( λ+ ∑ j∈ [ m ] λjρ ∗ j ) 2 . As a result , lim n , di→∞ di/n→zi Vλ , d , n = σ 2 m∑ i=1 λ2i ρ∗i + ∑ j∈ [ m ] λjJji ( λ+ ∑ j∈ [ m ] λjρ ∗ j ) 2 = σ2 ( λ 2 ) > ( ρ∗ + J > λ ) ( λ+ λ > ρ∗ ) 2 . By Lemma 12 , the variance is given by lim n , di→∞ di/n→zi Vλ , d , n = lim n , di→∞ di/n→zi −σ2 m∑ i=1 λ2iE [ ∂ ∂λi tr ( N−1 ) ] = σ2 m∑ i=1 r∗2i , where r∗ solves sup rt > 0 inf r1 , ... , rm > 0 2rt√1 + ∑ i∈ [ m ] r2i − 2rt ∑ i∈ [ m ] √ ziri + ∑ i∈ [ m ] 1 λi r2i − λr2t . Since { Vλ , d , n } is uniformly bounded and uniformly equicontinuous with respect to λ ∈ ( 0 , 1 ] by Lemma 5 , { Vλ , d , n } can be extended continuously to [ 0 , 1 ] and the family of extended functions is still uniformly bounded and uniformly equicontinuous . By the Arzela-Ascoli theorem , { Vλ , d , n } converges uniformly to the limit . By the Moore-Osgood theorem , we can exchange the two limits limn , di→∞ di/n→zi and limλ→0+ and get lim n , di→∞ di/n→zi lim λ→0+ Vλ , d , n = lim λ→0+ lim n , di→∞ di/n→zi Vλ , d , n = σ 2 m∑ i=1 r∗2i |λ=0 . E PROOF OF THEOREM 2 We use Theorem 1 to prove Theorem 2 . As in Theorem 1 , let r∗ solve minri≥0 maxrt≥0 ϑ ( rt , r , λ ) , where ϑ is defined in Equation ( 7 ) . Note that ϑ is a quadratic function of rt . Define A =√∑ i∈ [ m ] r 2 i + 1 , B = ∑ i∈ [ m ] √ ziri , A∗ = √∑ i∈ [ m ] r ∗2 i + 1 , and B ∗ = ∑ i∈ [ m ] √ zir ∗ i . Then r∗t = A−B λ and we get min ri≥0 max rt≥0 ϑ ( rt , r , λ ) = min ri≥0 ( A−B ) 2 λ + ∑ i∈ [ m ] 1 λi r2i = min ri≥0 ( A−B ) 2 λ + ∑ i∈ [ m ] 1 λi r2i . Taking the partial derivative with respect to ri gives ∂ ∂ri ( A−B ) 2 λ + ∑ i∈ [ m ] 1 λi r2i = 2 · A−B λ ( ri A − √ zi ) + 2 · ri λi . Setting it to zero gives the optimality condition for r∗i : A∗ −B∗ λ ( r∗i A∗ − √ zi ) = −r ∗ i λi , i ∈ [ m ] . ( 48 ) It follows that r∗i A∗ − √ zi r∗j A∗ − √ zj = r∗i /λi r∗j /λj , i , j ∈ [ m ] . Some algebraic manipulation in the above equation yields r∗i r∗j = λi λj · √ ziA ∗ − r∗i√ zjA∗ − r∗j , i , j ∈ [ m ] . Define z = ( z1 , . . . , zm ) . Then ‖z‖1 = ∑ i∈ [ m ] zi . By Cauchy–Schwarz inequality , if d/n →∑ i∈ [ m ] zi < 1 B ≤ √∑ i∈ [ m ] zi ‖r‖2 < ‖r‖2 < √ ‖r‖22 + 1 = A . Thus there does not exist r such that A = B . If d/n → ∑ i∈ [ m ] zi > 1 , then A = B is feasible for r. For example , set r = 1√ ( ‖z‖1 − 1 ) ‖z‖1 √ z . We have B = 〈 r , √ z 〉 = √ ‖z‖1 ‖z‖1 − 1 A = √ 1 + ‖r‖22 = √ 1 + 1 ‖z‖1 − 1 = B . If ‖z‖1 > 1 , since A = B is feasible , then lim λ→0+ min ri≥0 ( A−B ) 2 λ + ∑ i∈ [ m ] 1 λi r2i = min ri≥0 A=B ∑ i∈ [ m ] 1 λi r2i . If ‖z‖1 < 1 , then A−B always holds . To be precise , we have A−B ≥ ( √ ‖r‖22 + 1− ‖r‖2 ) ∨ ( ( 1− √ ‖z‖1 ) ‖r‖2 ) . If ‖r‖2 > 1 , then ( 1− √ ‖z‖1 ) ‖r‖2 > 1− √ ‖z‖1 . If ‖r‖2 ≤ 1 , then √ ‖r‖22 + 1−‖r‖2 ≥ √ 2−1 . Thus there exists a universal constant C0 = ( 1− √ ‖z‖1 ) ∨ ( √ 2− 1 ) > 0 such that A−B ≥ C0 . Recall Equation ( 48 ) . We have ( A∗ −B∗ ) ( r∗i A∗ − √ zi ) = −λr ∗ i λi , i ∈ [ m ] . Taking limλ→0+ , since A∗ −B∗ ≥ C0 does not go to zero , we have r∗i A∗ − √ zi = 0 , i ∈ [ m ] . Then we get r∗2i 1 + ∑ j∈ [ m ] r ∗2 j = zi , i ∈ [ m ] . Summing all i ∈ [ m ] yields ‖z‖1 = ∑ i∈ [ m ] r ∗2 i 1 + ∑ i∈ [ m ] r ∗2 i . Therefore , we have lim n , di→+∞ di/n→zi V0 , d , n = σ 2 lim λ→0+ m∑ i=1 r∗2i = σ 2 ∑ i∈ [ m ] zi 1− ∑ i∈ [ m ] zi . F PROOF OF THEOREM 3 Define A∗ = √∑ i∈ [ m ] r ∗2 i + 1 and B ∗ = ∑ i∈ [ m ] √ zir ∗ i . Equation ( 15 ) in Theorem 2 yields r∗1 r∗2 = λ1 λ2 · √ z1A ∗ − r∗1√ z2A∗ − r∗2 . Using the constraint A∗ = B∗ , we get r∗1 r∗2 = λ1 ( √ z1B ∗ − r∗1 ) λ2 ( √ z2B∗ − r∗2 ) . Define q = r ∗ 1 r∗2 . We have the following equation q = λ1 ( q ( z1 − 1 ) + √ z1z2 ) λ2 ( q √ z1z2 + z2 − 1 ) . Solving the above equation yields q = λ1 ( z1 − 1 ) + λ2 ( 1− z2 ) + √ ( λ1 ( z1 − 1 ) + λ2 ( 1− z2 ) ) 2 + 4λ1λ2z1z2 2λ2 √ z1z2 . ( 49 ) Here we discard the negative root . Let x = r∗21 + r ∗2 2 = r ∗2 2 ( 1 + q2 ) . ? ? yields 1 + x = r∗22 ( q √ z1 + √ z2 ) 2 = x 1 + q2 ( q √ z1 + √ z2 ) 2 . Solving x from the above equation gives x = q2 + 1 q2 ( z1 − 1 ) + 2q √ z1z2 + z2 − 1 . Therefore , lim n , di→+∞ di/n→zi V0 , d , n = q2 + 1 q2 ( z1 − 1 ) + 2q √ z1z2 + z2 − 1 . G PROOF OF THEOREM 4 Instead of considering the θ∗ specified in Equation ( 18 ) , we first consider a Bayesian setting where θ∗ ∼ N ( 0 , 1dId ) . Later , we will show that the setup in Equation ( 18 ) is asymptotically ( as di →∞ ) equivalent to this Bayesian setting . The precise meaning of equivalence will also be presented later . Our strategy can be divided into two steps . The first step is to show that the Bayes risk of the Bayes estimator is monotonically decreasing in the sample size n. The second step is to translate the sample-wise monotonicity of the Bayes estimator to the excess risk of the optimally regularized estimator θ̂λ , d , n in the setup of Equation ( 18 ) . Recall that since we are interested in sample-wise monotonicity , we add a subscript n to X and y ( they are defined by Equation ( 1 ) in Section 1.1 ) to emphasize that they consist of n data items . In this Bayesian setting , the likelihood function of θ∗ is L ( θ∗ | Xn , yn ) = ∏ i∈ [ n ] L ( θ∗ | xi , yi ) ∝ exp ( − ∑ i∈ [ n ] ( yi − 〈θ∗ , xi〉 ) 2 2σ2 ) = exp ( − ‖Xnθ∗ − yn‖22 2σ2 ) . The density of the prior of θ∗ is proportional to exp ( −d2 ‖θ ∗‖22 ) . Therefore , the posterior density of θ∗ is given by p ( θ∗ | Xn , yn ) ∝ exp ( − d ‖θ∗‖22 2 − ‖Xnθ∗ − yn‖22 2σ2 ) . As a result , the posterior distribution of θ∗ is Gaussian . The Bayes estimator is θ̂Bayes ( Xn , yn ) = arg min θ Eθ∗∼p ( θ∗|Xn , yn ) ‖θ − θ ∗‖2Σ . Taking the derivative with respect to θ gives ∂ ∂θ Eθ∗∼p ( θ∗|Xn , yn ) ‖θ − θ ∗‖2Σ = 2Σ ( θ − θ ∗ ) . Setting the above equation to zero yields Σ ( θ̂Bayes ( Xn , yn ) − Eθ∗∼p ( θ∗|Xn , yn ) θ∗ ) = 0 and there- fore θ̂Bayes ( Xn , yn ) = Eθ∗∼p ( θ∗|Xn , yn ) θ ∗ = E [ θ∗ | Xn , yn ] = arg min θ ( d ‖θ∗‖22 + ‖Xnθ∗ − yn‖22 σ2 ) . The final equality is because the posterior mean of a Gaussian distribution equals its mode . Define the Bayes risk Rn , Eθ∗∼N ( 0 , 1d Id ) , Xn , yn [ ∥∥∥θ̂Bayes ( Xn , yn ) − θ∗∥∥∥2 Σ ] . Write X = Rd and Y = R. Define R′n , inf θ̂ : Xn×Yn→R Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂ ( Xn , yn ) − θ∗∥∥∥2 Σ . We have R′n = inf θ̂ : Xn×Yn→Rd Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥Σ1/2θ̂ ( Xn , yn ) − Σ1/2θ∗∥∥∥2 2 = inf θ̂ : Xn×Yn→Rd Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂ ( Xn , yn ) − Σ1/2θ∗∥∥∥2 2 = Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥E [ Σ1/2θ∗ | Xn , yn ] − Σ1/2θ∗∥∥∥2 2 = Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥Σ1/2θ̂Bayes ( Xn , yn ) − Σ1/2θ∗∥∥∥2 2 = Rn . where the third equality is because the conditional expectation minimizes the ` 2 loss . Next , we want to show that Rn+1 ≤ Rn , i.e. , the Bayes risk of the Bayes estimator is monotonically decreasing in the sample size n. Rn+1 = inf θ̂ : Xn+1×Yn+1→Rd Eθ∗∼N ( 0 , 1d Id ) , Xn+1 , yn+1 [ ∥∥∥θ̂ ( Xn+1 , yn+1 ) − θ∗∥∥∥2 Σ ] ≤ inf θ̂ : Xn×Yn→R Eθ∗∼N ( 0 , 1d Id ) , Xn+1 , yn+1 [ ∥∥∥θ̂ ( Xn , yn ) − θ∗∥∥∥2 Σ ] = Rn . Then we want to show that Rn equals the Bayes risk of the optimally regularized estimator θ̂λ , n , d : Rn = inf λ≥0 Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ . Since Rn = inf θ̂ : Xn×Yn→Rd Eθ∗ , Xn , yn ∥∥∥θ̂ ( Xn , yn ) − θ∗∥∥∥2 Σ , we get Rn ≤ inf λ≥0 Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ . On the other hand , recalling θ̂Bayes ( Xn , yn ) = arg minθ ( d ‖θ‖22 + ‖Xnθ−yn‖22 σ2 ) = θ̂σ2d n , n , d and Rn = Eθ∗∼N ( 0 , 1d Id ) , Xn , yn [ ∥∥∥θ̂Bayes ( Xn , yn ) − θ∗∥∥∥2 Σ ] , we deduce Rn ≥ inf λ≥0 Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ . Therefore we deduce Rn = infλ≥0 Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ . As a result , we establish sample-wise monotonicity of the Bayes risk of optimal regularized θ̂λ , n , d : Rn+1 = inf λ≥0 Eθ∗ , Xn+1 , yn+1 ∥∥∥θ̂λ , n+1 , d − θ∗∥∥∥2 Σ ≤ inf λ≥0 Eθ∗ , Xn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ = Rn . ( 50 ) In what follows , we show that if θ∗ is given by Equation ( 18 ) , the excess risk of θ̂λ , n , d is asymptotically equal to its Bayes risk when θ∗ ∼ N ( 0 , 1dId ) : lim di→∞ ∣∣∣∣EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2Σ − Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2Σ ∣∣∣∣ = 0 . We abuse the notation in the above equation . The θ∗ in EXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2 Σ satisfies Equation ( 18 ) , while the θ∗ in Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ follows a normal distribution N ( 0 , 1dId ) . By Lemma 5 and Lemma 6 , if Σ = PΛP > and θ′ = P > θ∗ are as defined in Table 1 ( where P is an orthogonal matrix and Λ = diag ( λ1Id1 , . . . , λmIdm ) ∈ Rd×d is a diagonal matrix ) , for fixed θ∗ we have EXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2 Σ =EXn , yn [ ‖Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 θ′‖22 ] + σ2EXn , yn [ ‖ΛZ ( λnIn + Z > ΛZ ) −1 ‖22 ] , where every entry ofZ ∈ Rd×n follows i.i.d . N ( 0 , 1 ) . If θ∗ ∼ N ( 0 , 1dId ) , we have θ ′ ∼ N ( 0 , 1dId ) . Since the variance term σ2EXn , yn [ ‖ΛZ ( λnIn + Z > ΛZ ) −1 ‖22 ] does not depend on θ∗ , the two variance terms cancel out and we get EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ − Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ =EXn , yn [ ‖Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 θ′‖22 ] − EXn , yn , θ′∼N ( 0 , 1d Id ) [ ‖Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 θ′‖22 ] For U ∼ ⊕ i∈ [ m ] Unif ( O ( di ) ) , we have ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 d = ( Id + 1 nλ Λ1/2UZZ > U > Λ1/2 ) −1 = U ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 U > . By Lemma 13 , for θ∗ ( and thereby θ′ ) specified in Equation ( 18 ) , we get EXn , yn [ ‖Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 θ′‖22 ] = EXn , yn , φ [ ‖Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 φ‖22 ] , where φ ∼ ⊕ i∈ [ m ] Unif ( Sdi−1 ( ‖θ′i‖2 ) ) = ⊕ i∈ [ m ] Unif ( Sdi−1 ( √ di/d ) ) . In the Bayesian setting , if θ′ ∼ N ( 0 , 1dId ) , then U > θ′ ∼ N ( 0 , 1dId ) . We have Eθ′∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2 Σ = Eθ′∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥∥∥Λ1/2U ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 U > θ′ ∥∥∥∥∥ 2 2 = EXn , yn , ψ ∥∥∥∥∥Λ1/2 ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1 ψ ∥∥∥∥∥ 2 2 , where ψ = U > θ′ ∼ N ( 0 , 1dId ) . Next , we want to couple φ and ψ . Let si i.i.d.∼ Unif ( Sdi−1 ( 1 ) ) , hi i.i.d.∼ χ2 ( di ) , and define φ = √ d1/ds1 ... √ dm/dsm , ψ = √ h1/ds1 ... √ hm/dsm . We have ‖φ‖2 = 1 and ‖ψ‖2 = √√√√ m∑ i=1 hi d = √√√√ m∑ i=1 di d · hi di , ‖φ− ψ‖2 = √√√√ m∑ i=1 di d ( 1− √ hi di ) 2 . By the strong law of large numbers , limdi→+∞ hi/di = 1 almost surely . Thus we get limdi→+∞ , di/d→νi ‖ψ‖2 = √∑m i=1 νi and limdi→+∞ , di/d→νi ‖φ− ψ‖2 = 0 almost surely ( recall that we will let di → +∞ and di/d → νi for some constant νi > 0 . ) . Because∥∥Λ1/2∥∥ 2 . 1 and ∥∥∥ ( Id + 1nλΛ1/2ZZ > Λ1/2 ) −1∥∥∥ 2 ≤ ‖Id‖2 = 1 , we bound the norm of Q , Λ1/2 ( Id + 1 nλΛ 1/2ZZ > Λ1/2 ) −1 as follows ‖Q‖2 ≤ ∥∥∥Λ1/2∥∥∥ 2 ∥∥∥∥∥ ( Id + 1 nλ Λ1/2ZZ > Λ1/2 ) −1∥∥∥∥∥ 2 . 1 . It follows that ∣∣∣EXn , yn , φ [ ‖Qφ‖22 ] − EXn , yn , ψ [ ‖Qψ‖22 ] ∣∣∣ ≤EXn , yn , φ , ψ ∣∣∣‖Qφ‖22 − ‖Qψ‖22∣∣∣ =EXn , ynφ , ψ ( ‖Qφ‖2 + ‖Qψ‖2 ) |‖Qφ‖2 − ‖Qψ‖2| .EXn , ynφ , ψ [ ( ‖φ‖2 + ‖ψ‖2 ) ‖Q ( φ− ψ ) ‖2 ] .EXn , ynφ , ψ ‖φ− ψ‖2 , where the last inequality is because ‖φ‖2 + ‖ψ‖2 . 1 for all sufficiently large di . We know that limdi→+∞ , di/d→νi ‖φ− ψ‖2 = 0 almost surely . To apply Lebesgue ’ s dominated convergence theorem , we need to find a dominating integrable random variable . In fact , 1 + ‖ψ‖2 dominates ‖φ− ψ‖2 : ‖φ− ψ‖2 ≤ ‖φ‖2 + ‖ψ‖2 = 1 + ‖ψ‖2 . It is integrable because E ‖ψ‖2 = E [ √ χ2 ( d ) d ] ≤ √ E [ χ2 ( d ) ] d = 1 . Application of Lebesgue ’ s dominated convergence theorem yields lim di→+∞ , di/d→νi ∣∣∣EXn , yn , φ [ ‖Qφ‖22 ] − EXn , yn , ψ [ ‖Qψ‖22 ] ∣∣∣ = 0 . Therefore , we conclude that lim di→+∞ , di/d→νi ∣∣∣∣EXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2Σ − Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2Σ ∣∣∣∣ = 0 and this convergence is uniform in n and λ ∈ ( 0 , ∞ ) . It follows that lim n , di→∞ n/di→γi ∣∣∣∣EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ − Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ ∣∣∣∣ = 0 and this convergence is uniform in λ ∈ ( 0 , ∞ ) . By Lemma 8 ( the proof is similar when we replace α→ +∞ by n , di →∞ , n/di → γi ) , we have lim n , di→∞ n/di→γi ∣∣∣∣ infλ > 0EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2Σ − infλ > 0Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2Σ ∣∣∣∣ = 0 . ( 51 ) Define fα ( λ ) = Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ . We use α to denote the indices n , di . By Lemma 5 and Lemma 6 , we have Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ . Eθ∗∼N ( 0 , 1d Id ) ‖θ ∗‖22 = 1 . Therefore { fα ( λ ) } is uniformly bounded for λ > 0 . Since ∣∣∣∣ ddλEXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2 Σ ∣∣∣∣ . ‖θ∗‖22 and Eθ∗∼N ( 0 , 1d Id ) ‖θ ∗‖22 = 1 , we have∣∣∣∣ ddλEθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2Σ ∣∣∣∣ = ∣∣∣∣Eθ∗∼N ( 0 , 1d Id ) ddλEXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2Σ ∣∣∣∣ . 1 . As a result , { fα ( λ ) } is uniformly equicontinuous for λ > 0 , and in particular λ ∈ ( 0 , M ] for any M > 0 . Therefore { fα ( λ ) } can be extended continuously to [ 0 , M ] and the family of extended functions is still uniformly bounded and uniformly equicontinuous . Recall that if θ∗ ∼ N ( 0 , 1dId ) , we have θ′ ∼ N ( 0 , 1dId ) . As in Equation ( 19 ) , write θ ′ in a row-partitioned form θ′ = θ ′ 1 ... θ′m , where θ′i ∈ Rdi . Then ‖Πiθ∗‖2 = ‖θ′i‖2 ∼ √ χ2 ( di ) d = √ χ2 ( di ) di · did → √ νi as n , di → +∞ and n/di → γi , where νi = ( γi ∑ j∈ [ m ] 1 γj ) −1 . By Theorem 1 , { fα ( λ ) } converges pointwise , say , to h ( λ , γ1 , . . . , γm ) . By the Arzela-Ascoli theorem , limα fα ( λ ) = h ( λ , γ1 , . . . , γm ) uniformly on λ ∈ [ 0 , M ] . Therefore , as n , di →∞ and n/di → γi , by Lemma 8 , we have inf λ∈ [ 0 , M ] Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2 Σ → inf λ∈ [ 0 , M ] h ( λ , γ1 , . . . , γm ) . Recalling θ̂Bayes ( Xn , yn ) = arg minθ ( d ‖θ‖22 + ‖Xnθ−yn‖22 σ2 ) = θ̂σ2d n , n and Rn = Eθ∗∼N ( 0 , 1d Id ) , Xn , yn [ ∥∥∥θ̂Bayes ( Xn , yn ) − θ∗∥∥∥2 Σ ] = inf λ≥0 Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ . For all M > CM : = 2σ2 ∑ i∈ [ m ] 1 γi ≥ σ 2d n ( recall d n → ∑ i∈ [ m ] 1 γi ) , we have inf λ∈ [ 0 , M ] Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ = inf λ≥0 Eθ∗∼N ( 0 , 1d Id ) EXn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ → inf λ∈ [ 0 , M ] h ( λ , γ1 , . . . , γm ) . The uniqueness of limits implies that infλ∈ [ 0 , M ] h ( λ , γ1 , . . . , γm ) is independent of M as long as M > σ2 . As a result , if M > CM , we have infλ∈ [ 0 , M ] h ( λ , γ1 , . . . , γm ) = infλ≥0 h ( λ , γ1 , . . . , γm ) , which yields inf λ≥0 Eθ∗∼N ( 0 , 1d Id ) , Xn , yn ∥∥∥θ̂λ , n , d − θ∗∥∥∥2 Σ → inf λ≥0 h ( λ , γ1 , . . . , γm ) . ( 52 ) Equation ( 50 ) implies infλ≥0 h ( λ , γ1 , . . . , γm ) is decreasing in every γi . Combining Equation ( 51 ) and Equation ( 52 ) gives inf λ≥0 EXn , yn ∥∥∥θ̂λ , n − θ∗∥∥∥2 Σ → inf λ≥0 h ( λ , γ1 , . . . , γm ) .

Summary: This paper studies the generalization risk of ridge and ridgeless linear regression under Gaussian data and Gaussian noise settings. The limiting bias and variance are described in closed-form. The analysis leverages random matrix theory and CGMT. The authors show that sample-wise multiple descent phenomenon can take place or be avoided if optimal regularization is applied.