Title: Bayes beats Cross Validation: Efficient and Accurate Ridge Regression via Expectation Maximization
Venue: NeurIPS 2023
Accepted: yes
Keywords: Ridge Regression

Cross validation

Expectation Maximisation

Bayesian methods
Abstract: We present a novel method for tuning the regularization hyper-parameter, $\lambda$, of a ridge regression that is faster to compute than leave-one-out cross-validation (LOOCV) while yielding estimates of the regression parameters of equal, or particularly in the setting of sparse covariates, superior quality to those obtained by minimising the LOOCV risk. The LOOCV risk can suffer from multiple and bad local minima for finite $n$ and thus requires the specification of a set of candidate $\lambda$, which can fail to provide good solutions. In contrast, we show that the proposed method is guaranteed to find a unique optimal solution for large enough $n$, under relatively mild conditions, without requiring the specification of any difficult to determine hyper-parameters. This is based on a Bayesian formulation of ridge regression that we prove to have a unimodal posterior for large enough $n$, allowing for  both the optimal $\lambda$ and the regression coefficients to be jointly learned within an iterative expectation maximization (EM) procedure. Importantly, we show that by utilizing an appropriate preprocessing step, a single iteration of the main EM loop can be implemented in $O(\min(n, p))$ operations, for input data with $n$ rows and $p$ columns. In contrast, evaluating a single value of $\lambda$ using fast LOOCV costs $O(n \min(n, p))$ operations when using the same preprocessing. This advantage amounts to an asymptotic improvement of a factor of $l$ for $l$ candidate values for $\lambda$ (in the regime $q, p \in O(\sqrt{n})$ where $q$ is the number of regression targets).

Bayes beats Cross Validation: Fast and Accurate Ridge Regression via Expectation Maximization

Abstract

We present a novel method for tuning the regularization hyper-parameter, λ , of a ridge regression that is faster to compute than leave-one-out cross-validation (LOOCV) while yielding estimates of the regression parameters of equal, or particularly in the setting of sparse covariates, superior quality to those obtained by minimising the LOOCV risk. The LOOCV risk can suffer from multiple and bad local minima for finite n and thus requires the specification of a set of candidate λ , which can fail to provide good solutions. In contrast, we show that the proposed method is guaranteed to find a unique optimal solution for large enough n , under relatively mild conditions, without requiring the specification of any difficult to determine hyper-parameters. This is based on a Bayesian formulation of ridge regression that we prove to have a unimodal posterior for large enough n , allowing for both the optimal λ and the regression coefficients to be jointly learned within an iterative expectation maximization (EM) procedure. Importantly, we show that by utilizing an appropriate preprocessing step, a single iteration of the main EM loop can be implemented in O (min( n, p )) operations, for input data with n rows and p columns. In contrast, evaluating a single value of λ using fast LOOCV costs O ( n min( n, p )) operations when using the same preprocessing. This advantage amounts to an asymptotic improvement of a factor of l for l candidate values for λ (in the regime q, p ∈ O ( √ n ) where q is the number of regression targets).

1 Introduction

Ridge regression [25] is one of the most widely used statistical learning algorithms. Given training data X ∈ R n × p and y ∈ R n , ridge regression finds the linear regression coefficients ˆ β λ that minimize the ℓ 2 -regularized sum of squared errors, i.e.,

In practice, using ridge regression additionally involves estimating the value for the tuning parameter λ that minimizes the expected squared error E ( x T ˆ β λ -y ) 2 for new data x and y sampled from the same distribution as the training data. This problem is usually approached via the leave-one-out cross-validation (LOOCV) estimator, which can be computed efficiently by exploiting a closed-form solution for the leave-one-out test errors for a given λ . The wide and long-lasting use of the LOOCV approach suggests that it solves the ridge regression problem more or less optimally, both in terms of its statistical performance, as well as its computational complexity.

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However, in this work, we show that LOOCV is outperformed by a simple expectation maximization (EM) approach based on a Bayesian formulation of ridge regression. While the two procedures are not equivalent, in the sense that they generally do not produce identical parameter values, the EM estimates tend to be of equal quality or, particularly in sparse regimes, superior to the LOOCV estimates (see Figure 1). Specifically, the LOOCV risk estimates can suffer from potential multiple and bad local minima when using iterative optimization, or misspecified candidates when performing grid search. In contrast, we show that the EM algorithm finds a unique optimal solution for large enough n (outside pathological cases) without requiring any hard to specify hyper-parameters, which is a consequence of a more general bound on n (Thm. 3.1) that we establish to guarantee the unimodality of the posterior distribution of Bayesian ridge regression-a result with potentially wider applications. In addition, the EM procedure is asymptotically faster than the LOOCV procedure by a factor of l where l is the number of candidate values for λ to be evaluated (in the regime p, q ∈ O ( √ n ) where p , q , and n are the number of covariates, target variables, and data points, respectively). In practice, even in the usual case of q = 1 and l = O (1) , the EM algorithm tends to outperform LOOCV computationally by an order of magnitude as we demonstrate on a test suite of datasets from the UCI machine learning repository and the UCR time series classification archive.

While the EM procedure discussed in this paper is based on a recently published procedure for learning sparse linear regression models [44], the adaption of this procedure to ridge regression has not been previously discussed in the literature. Furthermore, a direct adoption would lead to a main loop complexity of O ( p 3 ) that is uncompetitive with LOOCV. Therefore, in addition to evaluating the empirical accuracy and efficiency of the EM algorithm for ridge regression, the main technical contributions of this work are to show how certain key quantities can be efficiently computed from either a singular value decomposition of the design matrix, when p ≥ n , or an eigenvalue decomposition of the Gram matrix X T X , when n > p . This results in an E-step of the algorithm in time O ( r ) where r = min( n, p ) , and an M-step found in closed form and solved in time O (1) , yielding an ultra-fast main loop for the EM algorithm.

These computational advantages result in an algorithm that is computationally superior to efficient, optimized implementations of the fast LOOCV algorithm. Our particular implementation of LOOCV actually outperforms the implementation in scikit-learn by approximately a factor of two by utilizing a similar preprocessing to the EM approach. This enables an O ( nr ) evaluation for a single λ (which is still slower than the O ( r ) evaluation for our new EM algorithm; see Table 1 for an overview of asymptotic complexities of LOOCV and EM), and may be of interest to readers by itself. Our implementation of both algorithms, along with all experiment code, are publicly available in the standard package ecosystems of the R and Python platforms, as well as on GitHub 1 .

In the remainder of this paper, we first briefly survey the literature of ridge regression with an emphasis on the use of cross validation (Sec. 2). Based on the Bayesian interpretation of ridge regression, we then introduce the EM algorithm and discuss its convergence (Sec. 3). Finally, we develop fast implementations of both the EM algorithm and LOOCV (Sec. 4) and compare them empirically (Sec. 5).

2 Ridge Regression and Cross Validation

Ridge regression [25] (also known as ℓ 2 -regularization) is a popular method for estimation and prediction in linear models. The ridge regression estimates are the solutions to the penalized leastsquares problem given in (1). The solution to this optimization problem is given by:

When λ → 0 , the ridge estimates coincide with the minimum ℓ 2 norm least squares solution [22, 31], which simplifies to the usual least squares estimator in cases where the design matrix X has full column rank (i.e. X T X is invertible). Conversely, as λ → ∞ , the amount of shrinkage induced by the penalty increases, with the resulting ridge estimates becoming smaller for larger values of λ . Under fairly general assumptions [26], including misspecified models and random covariates of growing dimension, the ridge estimator is consistent and enjoys finite sample risk bounds for all fixed λ ≥ 0 , i.e., it converges almost surely to the prediction risk minimizer, and its squared deviation from this minimizer is bounded for finite n with high probability. However, its performance can still vary greatly with the choice of λ ; hence, there is a need to estimate the optimal value from the given training data.

Earlier approaches to this problem [e.g. 2, 7, 14, 19, 23, 28-30, 34, 48] rely on an explicit estimate of the (assumed homoskedastic) noise variance, following the original idea of Hoerl and Kennard [25]. However, estimating the noise variance can be problematic, especially when p is not much smaller than n [18, 20, 24, 46]. More recent approaches adopt model selection criteria to select the optimal λ without requiring prior knowledge or estimation of the noise variance. These methods involve minimizing a selection criterion of choice, such as the Akaike information criterion (AIC) [1], Bayesian information criterion (BIC) [40], Mallow's conceptual prediction (C p ) criterion [33], and, most commonly, cross validation (CV) [4, 49].

A particularly attractive variant of CV is leave-one-out cross validation (LOOCV), also referred to as the prediction error sum of squares (PRESS) statistic in the statistics literature [3]

where ˆ y i = ˜ x i ˆ β -i λ , and ˆ β -i λ denotes the solution to (2) when the i -th data point (˜ x i , y i ) is omitted. LOOCV offers several advantages over alternatives such as 10-fold CV: it is deterministic, nearly unbiased [47], and there exists an efficient "shortcut" formula for the LOOCV ridge estimate [36]:

where H ( λ ) = X ( X T X + λ I p ) -1 X T is the regularized 'hat', or projection, matrix and e = y -H ( λ ) y are the residuals of the ridge fit using all n data points. As it only requires the diagonal entries of the hat matrix, Eq. (4) allows for the computation of the PRESS statistic with the same time complexity O ( p 3 + np 2 ) as a single ridge regression fit.

Moreover, unless p/n → 1 , the LOOCV ridge regression risk as a function of λ converges uniformly (almost surely) to the true risk function on [0 , ∞ ) and therefore optimizing it consistently estimates the optimal λ [22, 36]. However, for finite n , the LOOCV risk can be multimodal and, even worse, there can exist local minima that are almost as bad as the worst λ [42]. Therefore, iterative algorithms like gradient descent cannot be reliably used for the optimization, giving theoretical justification for the pre-dominant approach of optimizing over a finite grid of candidates L = ( λ 1 , . . . , λ l ) . Unfortunately, despite the true risk function being smooth and unimodal, a naïvely chosen finite grid cannot be guaranteed to contain any candidate with a risk value close to the optimum. While this might not pose a problem for small n when the error in estimating the true risk via LOOCV is likely large, it can potentially become a dominating source of error for growing n and p . Therefore, letting l grow moderately with the sample size appears necessary, turning it into a relevant factor in the asymptotic time complexity.

As a further disadvantage, LOOCV (or CV in general) is sensitive to sparse covariates, as illustrated in Figure 1 where the performance of LOOCV, relative to the proposed EM algorithm, degrades as the noise variance σ 2 grows. In the sparse covariate setting, a situation common in genomics, the information about each coefficient is concentrated in only a few observations. As LOOCV drops an observation to estimate future prediction error, the variance of the CV score can be very large when the predictor matrix is very sparse, as the estimates depend on only a small number of the remaining observations. In the most extreme case, known as the multiple means problem [27], X = I n , and all the information about each coefficient is concentrated in a single observation. In this setting, the LOOCV score reduces to ∑ y 2 i , and provides no information about how to select λ . In contrast, the proposed EM approach explicitly ties together the coefficients via the probabilistic Bayesian interpretation of λ as the inverse-variance of the unknown coefficient vector. This 'borrowing of strength' means that the procedure provides a sensible estimate of λ even in the case of multiple means (see Appendix A).

3 Bayesian Ridge Regression

The ridge estimator (2) has a well-known Bayesian interpretation; specifically, if we assume that the coefficients are a priori normally distributed with mean zero and common variance τ 2 σ 2 we obtain a Bayesian version of the usual ridge regression procedure, i.e.,

where π ( · ) is an appropriate prior distribution assigned to the variance hyperparameter τ 2 . For a given τ > 0 and σ > 0 , the conditional posterior distribution of β is also normal [32]

where the posterior mode (and mean) ˆ β τ is equivalent to the ridge estimate with penalty λ = 1 /τ 2 (we rely on the variable name in the notation ˆ β x to indicate whether it refers to (6) or (2)).

Shrinkage Prior To estimate the τ 2 hyperparameter in the Bayesian framework, we first must choose a prior distribution for the hypervariance τ 2 . We assume that no strong prior knowledge on the degree of shrinkage of the regression coefficients is available, and instead assign the recommended default beta-prime prior distribution for τ 2 [15, 38] with probability density function:

where B ( a, b ) is the beta function. Specifically, we choose a = b = 1 / 2 , which corresponds to a standard half-Cauchy prior on τ . The half-Cauchy is a heavy-tailed, weakly informative prior that is frequently recommended as a default choice for scale-type hyperparameters such as τ [38]. Further, this estimator is very insensitive to the choice of a or b . As demonstrated by Theorem 6.1 in [6], the marginal prior density over β , ∫ π ( β | τ 2 ) π ( τ 2 | a, b ) dτ 2 = π ( β | a, b ) has polynomial tails in ∥ β ∥ 2 for all a > 0 , b > 0 , and has Cauchy or heavier tails for b ≤ 1 / 2 . This type of polynomial-tailed prior distribution over the norm of the coefficients is insensitive to the overall scale of the coefficients, which is likely unknown a priori . This robustness is in contrast to other standard choices of prior distributions for τ 2 such as the inverse-gamma distribution [e.g., 35, 41] which are highly sensitive to the choice of hyperparameters [38].

Unimodality and Consistency The asymptotic properties of the posterior distributions in Gaussian linear models (5) have been extensively researched [8, 16, 17, 45]. These studies reveal that in linear models, the posterior distribution of β is consistent, and converges asymptotically to a normal distribution centered on the true parameter value. When p is fixed, this assertion can be established through the Bernstein-Von Mises theorem [45, Sec. 10.2]. Our specific problem (5) satisfies the conditions for this theorem to hold: 1) both the Gaussian-linear model p ( y | β , σ 2 ) and the marginal distribution ∫ p ( y | β , σ 2 ) π ( β | τ 2 ) d β = p ( y | τ 2 , σ 2 ) are identifiable; 2) they have well defined Fisher information matrices; and 3) the priors over β and τ are absolutely continuous. Further, these asymptotic properties remain valid when the number of predictors p n is allowed to grow with the sample size n at a sufficiently slower rate [8, 16].

The following theorem (see the proof in Appendix B) provides a simple bound on the number of samples required to guarantee that the posterior distribution for the Bayesian ridge regression hierarchy given by (5) has only one mode outside a small environment around zero.

Theorem 3.1. Let ϵ > 0 , and let γ n be the smallest eigenvalue of X T X /n . If γ n > 0 and ϵ > 4 / ( nγ n ) then the joint posterior p ( β , σ 2 , τ 2 | y ) has a unique mode with τ 2 ≥ ϵ . In particular, if γ n ≥ cn -α with α < 1 and c > 0 then there is a unique mode with τ 2 ≥ ϵ if n > (4 / ( cϵ )) 1 / (1 -α ) .

In other words, all sub-optimal non-zero posterior modes vanish for large enough n if the smallest eigenvalue of X T X grows at least proportionally to some positive power of n . This is a very mild assumption that is typically satisfied in fixed as well as random design settings, e.g., with high probability when the smallest marginal covariate variance is bounded away from zero.

̸

Expectation Maximization Given the restricted unimodality of the joint posterior (5) for large enough n , in conjunction with its asymptotic concentration around the optimal β 0 , estimating the model parameters via an EM algorithm appears attractive, as they are guaranteed to converge to an exact posterior mode. In particular, in the non-degenerate case that β 0 = 0 , there exist τ 2 = ϵ 2 > 0 , such that for large enough, but finite n , the posterior concentrates around ( β 0 , τ 2 ) , and thus β 0 is identified by EM if initialized with a large enough τ 2 .

Specifically, we use the novel approach [44] in which the coefficients β are treated as 'missing data', and τ 2 and σ 2 as parameters to be estimated. Given the hierarchy (5), the resulting Bayesian EM algorithm then solves for the posterior mode estimates of β by repeatedly iterating through the following two steps until convergence:

E-step . Find the parameters of the Q-function , i.e., the expected complete negative log-posterior (with respect to β ), conditional on the current estimates of ˆ τ 2 t and ˆ σ 2 t , and the observed data y :

where the quantities to be computed are the (conditionally) expected sum of squared errors ESS = E [ || y -X β || 2 | ˆ τ 2 t , ˆ σ 2 t ] and the expected squared norm ESN = E [ ∥ β ∥ 2 | ˆ τ 2 t , ˆ σ 2 t ] . Denoting by tr( · ) the trace operator, one can show (see Appendix C) that these quantities can be computed as

M-step . Update the parameter estimates by minimizing the Q-function with respect to the shrinkage hyperparameter τ 2 and noise variance σ 2 , i.e.,

Instead of numerically optimizing the two-dimensional Q-function (10), we can derive closed-form solutions for both parameters by first finding ˆ σ 2 ( τ 2 ) , i.e., the update for σ 2 , as a function of τ 2 , and then substituting this into the Q-function. This yields a Q-function that is no longer dependent on σ 2 , and solving for ˆ τ 2 is straightforward. The resulting parameter updates in the M-step are given by:

where g = (4 n +4)ESN(3 + p )ESS + ((1 -n )ESN + ( p +1)ESS) 2 . The derivations of these formulae are presented in Appendix D.

From (11), we see that updating the parameter estimates in the M-step requires only constant time. Therefore, the overall efficiency of the EM algorithm is determined by the computational complexity of the E-step. Computing the parameters of the Q-functions directly via (9) requires inverting A τ , resulting in O ( p 3 ) operations. In the next section, we show how to substantially improve this approach via singular value decomposition.

4 Fast Implementations via Singular Value Decomposition

To obtain efficient implementations of the E-Step of the EM algorithm as well as of the LOOCV shortcut formula, one can exploit the fact that the ridge solution is preserved under orthogonal transformations. Specifically, let r = min( n, p ) and m = max( n, p ) and let UΣV T = X be a compact singular value decomposition (SVD) of X . That is, U ∈ R n × r and V ∈ R p × r are semiorthonormal column matrices, i.e., U T U = I n and V T V = I p , and Σ = diag( s 1 , . . . , s r ) ∈ R r × r is a diagonal matrix that contains the non-zero singular values s = ( s 1 , . . . , s r ) of X . With this decomposition, and an additional O ( nr ) pre-processing step to compute c = ΣU T y , we can compute the ridge solution α τ ∈ R r for a given τ with respect to the rotated inputs XV in time O ( r ) via

where a ⊙ b denotes the element-wise Hadamard product of vectors a and b . The compact SVD itself can be obtained in time O ( mr 2 ) via an eigendecomposition of either X T X = VΣ 2 V T in case n ≥ p or XX T = UΣ 2 U T in case n < p followed by the computation of the missing U = XVΣ -1 or V = X T UΣ -1 .

In summary, after an O ( mr 2 ) pre-processing step, we can obtain rotated ridge solutions for an individual candidate τ in time O ( r ) . Moreover, for the optimal τ ∗ , we can find the ridge solution ˆ β τ ∗ = V ˆ α τ ∗ with respect to the original input matrix via an O ( pr ) post-processing step. Below we show how the key statistics that have to be computed per candidate τ (and σ ) can be computed efficiently based on ˆ α τ , the pre-computed c , and SVD. For the EM algorithm, these are the posterior squared norm and sum of squared errors, and for the LOOCV algorithm, this is the PRESS statistic. While the main focus of this work is the EM algorithm, the fast computation of the PRESS shortcut formula appears to be not widely known (e.g., the current implementation in both scikit-learn and glmnet do not use it) and may therefore be of independent interest.

ESN For the posterior expected squared norm ESN = σ 2 tr( A -1 τ ) + ∥ ˆ β τ ∥ 2 , we first observe that ∥ ˆ β τ ∥ 2 = ∥ V ˆ α τ ∥ 2 = ∥ ˆ α τ ∥ 2 , and then note that the trace can be computed as

where in the first equation we denote by V p , Σ 2 p the full matrices of eigenvectors and eigenvalues of X T X (including potential zeros), and in the second equation we used the cyclical property of the trace. Thus, all quantities required for ESN can be computed in time O ( r ) given the SVD and ˆ α τ .

ESS For the posterior expected sum of squared errors ESS = ∥ y -X ˆ β τ ∥ 2 + σ 2 tr( X T XA -1 τ ) , we can compute the residual sum of squares term via

where we use ˆ β τ = V ˆ α τ and X = UΣV T in the first equation and the orthonormality of U and the definition of c = ΣU T y in the second. Finally, for the trace term, we find that

PRESS The shortcut formula of the PRESS statistic (4) for a candidate λ requires the computation of the diagonal elements of the hat matrix H ( λ ) = X ( X T X + λ I p ) -1 X T as well as the residual vector e = y -Hy . With the SVD, the first simplifies to

where we use the fact that diagonal matrices commute. This allows to compute the desired diagonal elements h ii in time O ( r ) via

where u ij denotes the elements of U . Computing the residual vector is easily done via the rotated ridge solution e = y -UΣ ˆ α λ . However, this still requires O ( nr ) operations, simply because there are n residuals to compute.

Thus, in summary, by combining the pre-processing with the fast computation of the PRESS statistic, we obtain an overall O ( mr 2 + lqnr ) implementation of ridge regression via LOOCV where l denotes the number of candidate λ and q the number of regression target variables. In contrast, for the EM algorithm, by combining the fast computation of ESS and ESN, we end up with an overall complexity of O ( mr 2 + kqr ) where k denotes the number of EM iterations. If we further assume that k = o ( n ) , which is supported by experimental results, see Sec. 5, and that both q, p = O ( √ n ) there is an asymptotic advantage of a factor of l of the EM approach. This regime is common in settings where more data allows for more fine-grained input as well as output measurements, e.g., in satellite time series classification via multiple target regression [11, 37]. All time complexities are summarized in Tab. 1 and detailed pseudocode for both the fast EM algorithm and the fast LOOCV algorithm is provided in the Appendix (see Table 3 and 4).

5 Empirical Evaluation

In this section, we compare the predictive performance and computational cost of LOOCV against the proposed EM method. We present numerical results on both synthetic and real-world datasets. To implement the LOOCV estimator, we use a predefined grid, L = ( λ 1 , . . . , λ l ) . We use the two most common methods for this task: (i) fixed grid - arbitrarily selecting a very small value as λ min , a large value as λ max , and construct a sequence of l values from λ max to λ min on log scale; (ii) data-driven grid - find the smallest value of λ max that sets all the regression coefficient vector to zero 2 (i.e. ˆ β = 0 ), multiply this value by a ratio such that λ min = κλ max and create a sequence from

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λ max to λ min on log scale. The latter method is implemented in the glmnet package in combination with an adaptive κ coefficient

which we replicate here as input to our fast LOOCV algorithm (Appendix, Table 4) to efficiently recover the glmnet LOOCV ridge estimate. 3

We consider a fixed grid of λ = (10 -10 , . . . , 10 10 ) and the grid based on the glmnet heuristic; in both cases, we use a sequence of length 100. The latter is a data-driven grid, so we will have a different penalty grid for each simulated or real data set. Our EM algorithm does not require a predefined penalty grid, but it needs a convergence threshold which we set to be ϵ = 10 -8 . All experiments in this section are performed in Python and the R statistical platform. Datasets and code for the experimental results is publicly available. As is standard in penalized regression, and without any loss of generality, we standardized the data before model fitting. This means that the predictors are standardized to have zero mean, standard deviation of one, and the target has a mean of zero, i.e., the intercept estimate is simply ˆ β 0 = (1 /n ) ∑ y i .

5.1 Simulated Data

In this section, we use a simulation study to investigate the behavior of EM and LOOCV as a function of the sample size, n , and two other parameters of interest: the number of covariates p , and the noise level of the target variable. In particular, we are interested in the parameter estimation performance, the corresponding λ -values, and the computational cost. To gain further insights into the latter, the number of iterations performed by the EM algorithm is of particular interest, as we do not have quantitative bounds for its behavior. We consider two settings that vary in the level of sparsity and correlation structure of the covariates. The first setting (Fig. 1) assumes i.i.d Bernoulli distributed covariates with small success probabilities that result in sparse covariate matrices, while the second setting (Fig. 2) assumes normally distributed covariates with random non-zero covariances. In both cases, the target variable is conditionally normal with mean x T β for a random β drawn from a standard multivariate normal distribution.

Looking at the results, a common feature of both settings is that the computational complexity of the EM algorithm is a non-monotone function in n . In contrast to LOOCV, the behavior of EM shows distinctive phases where the complexity temporarily decreases with n before it settles into the, usually expected, monotonically increasing phase. As can be seen, this is due to the behavior of the number of iterations k , which peaks for small values of n before it declines rapidly to a small

constant (around 10) when the cost of the pre-processing begins to dominate. The occurrence of these phases is more pronounced for both growing p and growing σ . This behavior is likely due to the convergence to normality of the posterior distribution as the sample size n →∞ , with convergence being slower for large p .

An interesting observation is that CV with the employed glmnet grid heuristic fails, in the sense that the resulting ridge estimator does not appear to be consistent for large p in Setting 2. This is due to the minimum value of λ produced by the glmnet heuristic being too large, and the resulting ridge estimates being overshrunk. This clearly underlines the difficulty of choosing a robust dynamic grid a problem that our EM algorithm avoids completely.

5.2 Real Data

We evaluated our EM method on 24 real-world datasets. This includes 21 datasets from the UCI machine learning repository [5] (unless referenced otherwise) for normal linear regression tasks and 3 time-series datasets from the UCR repository [10] for multitarget regression tasks. The latter is a multilabel classification problem in which the feature matrix was generated by the state-of-the-art HYDRA [12] time series classification procedure (which by default uses LOOCV ridge regression for classification), and we train q ridge regression models in a one-versus-all fashion, where q is the number of target classes. The datasets were chosen such that they covered a wide range of sample sizes, n , and number of predictors, p . We compared our EM algorithm against the fast LOOCV in terms of predictive performance, measured in R 2 (and classification accuracy) on the test data, and computational efficiency.

Our linear regression experiments involve 3 settings: (i) standard linear regression; (ii) secondorder multivariate polynomial regression with added interactions and second-degree polynomial transformations of variables, and (iii) third-order multivariate polynomial regression with added three-way interactions and cubic polynomial transformations. For each experiment, we repeated the process 100 times and used a random 70/30 train-test split. Due to memory limitations, we limit our design matrix size to a maximum of 35 million entries. If the number of transformed predictors exceeded this limit, we uniformly sub-sampled the interaction variables to ensure that p ∗ ≤ 35000000 / (0 . 7 n ) , and then fit the model using the sampled variables. Note that we always keep the original variables (main effects) and sub-sampled the interactions. In the case of multitarget regression, we performed a random 70/30 train-test split and repeated the experiment 30 times. To ensure efficient reproducibility of our experiments, we set a maximum runtime of 3 hours for each dataset. Any settings that exceeded this time limit were consequently excluded from the result table.

Table 2 details the results of our experiments; specifically, the ratio of time taken to run fast LOOCV divided by the time taken to run our EM procedure ( T ), and the R 2 values obtained by both methods on the withheld test set. The number of features, p , and observations, n recorded are values after data preprocessing (missing observations removed, one-hot encoding transformation, etc.). The results demonstrate that our EM algorithm can be up to 49 times faster than the fast LOOCV, with the speed-ups becoming more apparent as the sample size n and the number of target variables q increases. In addition, we see that this advantage in speed does not come at a cost in predictive performance, as our EM approach is comparable to, if not better than, LOOCV in almost all cases (also see Appendix, Figure 1, in which most of R 2 values are distributed along the diagonal line).

An interesting observation is that LOOCV using the fixed grid can occasionally perform extremely poorly (as indicated by large negative R 2 values) while LOOCV using the glmnet grid does not seem to exhibit this behavior. This appears likely to be due to the grid chosen using the glmnet heuristic. Its performance is artificially improved because it is unable to evaluate sufficiently small values of λ and is not actually selecting the very small λ value that minimizes the LOOCV score. The incorrectly large λ values are providing protection in these examples from undershrinkage.

6 Conclusion

The introduced EM algorithm is a robust and computationally fast alternative to LOOCV for ridge regression. The unimodality of the posterior guarantees a robust behavior for finite n under mild conditions relative to LOOCV grid search, and the SVD preprocessing enables an overall faster computation with an ultra-fast O ( k min( n, p )) main loop. Combining this with a procedure such as orthogonal least-squares to provide a highly efficient forward selection procedure is a promising avenue for future research. As the Q-function is an expected negative log-posterior, it offers a score

1 This dataset is not from the UCI repository. Data references can be found in the appendix.

2 Time-series dataset from the UCR repository. EM, Fix, and GLM are the classification accuracy on the test data on which the usefulness of predictors themselves may be assessed, i.e., a model selection criteria, resulting in a potentially very accurate and fast procedure for sparse model learning. An important open problem is the theoretical analysis of the expected number of EM iterations k that is required for convergence. The empirical evidence suggests that k converges to a constant, and is thus negligible in the asymptotic time complexity. This is in alignment with the convergence of the posterior to a multivariate normal distribution. However, such intuitive and empirical arguments cannot replace a rigorous worst-case analysis.

Acknowledgments and Disclosure of Funding

We thank the anonymous reviewers for their valuable feedback that led to a substantially improved theoretical analysis. This work was supported by the Australian Research Council (DP210100045).

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A EMProcedure for Multiple Means

We show that the Bayesian EM procedure provides sensible estimates of the regularization parameter even in the setting of the normal multiple means problem with known variance σ 2 . In this setting, the LOOCV is unable to provide any guidance on how to choose λ due to all the information for each regression parameter being concentrated in a single observation. We use the τ parameterisation of the hyperparameter, rather than τ 2 , as the resulting estimator has an easy to analyse form.

In the normal multiple means model, we are given ( y i | β i ) ∼ N ( β i , 1) , i.e., y is a p -dimensional normally distributed vector with mean β and identity covariance matrix. The conditional posterior distribution of β is:

where κ = 1 / (1 + τ 2 ) . Under this setting, Strawderman [43] proved that if p ≥ 3 , then any estimator of the form

where 0 ≤ r ( 1 2 || y || 2 ) ≤ 2 and r ( · ) is non-decreasing, is minimax, i.e., it dominates least-squares. We will now show that our EM procedure not only yields reasonable estimates in this setting, in contrast to LOOCV, but that these estimates are minimax, and hence dominate least-sqaures.

For the normal means model, we can obtain a closed form solution for the optimum τ , by solving for the stationary point for which τ t +1 = τ t , with τ ∼ C + (0 , 1) :

with solution κ = ( p +2) /s . Plugging this κ solution into (17), we note that the resulting estimator of β (17) is of the form (18) with

As we have r ( 1 2 || y || 2 ) ≤ 2 when p ≥ 6 , the EM ridge estimator is minimax in this setting for p ≥ 6 .

B Proof of Theorem 3.1

We prove that for sufficiently large n , a continuous injective reparameterization of the negative log joint posterior of (5) & (7) is convex when restricted to τ 2 ≥ ϵ . This is sufficient, since unimodality is preserved by strictly monotone transformations and continuous injective reparameterizations.

Specifically, for the presented hierarchical model, the negative log joint posterior up to an additive constant is

and reparameterising with ϕ = β/σ , ρ = 1 /σ and χ = 1 /τ and reorganising terms yields

The first three terms can easily be checked to be convex via a second derivative test, for which the convexity of the second term is contingent on the condition that a < 1 + p/ 2 , a condition that holds true in our specific scenario with a = b = 1 / 2 . For the last two terms, the combined Hessian is of block form [ A , B ; B T , C ] with A = X T X + χ 2 I p , B = 2 χ ϕ , and C = ∥ ϕ ∥ 2 . Symmetric matrices of this form are positive definite if A and its Schur complement

are positive definite. Clearly, A is positive definite. Moreover, for n > 4 / ( ϵ 2 γ n ) , we have

where we used the fact that VV T is the orthogonal projection onto the column space of V . The overall inequality implies the required positivity of the Schur complement.

C Derivation of Equation 9

C.1 Derivation of ESN

Here we show that

This is a rather straightforward proof. We use the fact that given a random variable x ; the expected squared value of x is E [ x 2 ] = Var[ x ] + E[ x ] 2 .

C.2 Derivation of ESS

Here we show that

We first provide an important fact on the quadratic forms of random variables in Lemma C.1 below: Lemma C.1. Let b be a p-dimensional random vector and A be a p-dimensional symmetric matrix. If E[ b ] = µ and Var( b ) = Σ , then E [ b T Ab ] = tr( AΣ ) + µ T A µ .

Now, we expand the left-hand side of Equation 19 :

The use of lemma C.1 allows Equation 20 to be rewritten as

D Solving for the parameter updates (Derivation of Equation 11)

Rather than solving a two-dimensional numerical optimization problem (10), we show that given a fixed τ 2 , we can find a closed formed solution for σ 2 , and vice versa. To start off, we need to find the solution for σ 2 as a function of τ 2 . First, find the negative logarithm of the joint probability distribution of hierarchy (5):

Dropping terms that do not depend on σ 2 yields:

Solving the above minimization problem involves differentiating the negative logarithm with respect to σ 2 and solving for σ 2 that set the derivative to zero. This gives us:

Next, to obtain the M-step updates for the shrinkage parameter τ 2 , we repeat the same procedure find the negative logarithm of the joint probability distribution and remove terms that do not depend on either σ 2 or τ 2 :

Substiting the solution for σ 2 (23) into equation (24), yields a Q-function that depends only on τ 2 . We eliminate the dependency on σ 2 by finding the optimal σ 2 as a function of τ 2 and substitute it into the Q-function of (24):

Differentiating (25) with respect to τ 2 and solving for the τ 2 that set the derivative to zero yields:

The τ 2 update is the positive solution to the quadratic equation (in terms of τ 2 ) (26):

E Supplementary Results Material

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles the problem of choosing the regularization parameter in ridge regression. The authors introduce a Bayesian model that has the ridge estimate as its posterior mode with a prior distribution over the regularization parameter; they then propose an EM algorithm to estimate the optimal regularization parameter. The authors compare their estimator to leave-one-out CV (LOOCV), and note that each iteration of their EM algorithm is faster than evaluating LOOCV for one value of the regularization parameter. In experiments, the number of EM iterations is modest, so the author's algorithm is faster than LOOCV.\n\nSTRENGTHS:\nThis paper is well-written and easy to follow (maybe in the top 50% of NeurIPS papers in this regard). I thought the authors made their contributions very clear, and they have extensive experiments. I also think the authors did a good job of backing up all of their specific claims with either experiments or theory.\n\nWEAKNESSES:\n**Editing this here on 8-24-2023** because it seems I can't list authors as readers to any new replies at this point.\n\nThe global comment proving the unimodality of the EM objective is great! I'm convinced, and will raise my score. The strength that I see of this paper is that it is concretely improving on LOOCV in two ways: it has a runtime improvement per iteration (and solid empirical evidence that the number of iterations required for convergence in the EM algorithm is small), and it is guaranteed to be unimodal as soon as $n$ is modest compared to the smallest eigenvalue of the covariance matrix of the covariates. A couple remaining notes that I think could be good to address in a potential camera-ready:\n\n1. LOOCV can be guaranteed to have only one local minimum -- see Stephenson et al. (2021). But, compared to the conditions under which the current authors' EM objective is guaranteed to have one minimum, the conditions in Stephenson et al. (2021) are way more complicated, and their proof is way more complicated. And their experiments show that LOOCV will often have multiple local optima in normal conditions. I think these results make the authors' result in the global comment even stronger.\n2. I still think the paper could be made stronger by an empirical comparison to gradient descent on the LOOCV objective. But I think the paper is solid as-is with the inclusion of all the discussion above and the new proof in the global comment.\nBest, fBDz\n\nWilliam T. Stephenson, Zachary Frangella, Madeleine Udell, Tamara Broderick. Can we globally optimize cross-validation loss? Quasiconvexity in ridge regression. NeurIPS 2021.\n\n---------------------------------------------------------------\n\n\nI think a method that in both accuracy and computation beats LOOCV for ridge regression would be big news for the ML community. I'm not entirely convinced that the proposed method is doing that, though. And something that is big news also needs to come with a lot backing it up. First, a lot of the issues with LOOCV in the paper seem to stem from the choice of a finite grid over the regularization parameter -- the authors point out that it is slow to evaluate LOOCV for all points in the grid, and also the choice of grid can be tricky. It seems like the authors' method only gets around these issues by virtue of being an optimization algorithm. However, there is a much more common optimization algorithm: gradient descent over the LOOCV objective. True, the authors point out that the LOOCV objective can have local minima. But it often doesn't. How do we know the same isn't true of the authors' method? I really think a detailed comparison to this approach is needed, as gradient descent seems to have many of the same strengths as the authors' EM method and an equal number of weaknesses (i.e. I think we should assume both can have issues with local optima until proven otherwise).\n\nMy second main issue with the paper is that it doesn't really justify where this EM algorithm came from or what it's really doing. **Why** is this algorithm outperforming LOOCV? What is it picking up on that LOOCV is not? Why not use some other miscellaneous approach? E.g. could we sample from the posterior and take the posterior mean of $\\tau$? How do the chosen priors on $\\sigma$ and $\\tau$ affect things, and why are those good priors on $\\sigma$ and $\\tau$? Overall I this approach needs to be a lot more explained and justified if the proposal is to replace LOOCV with it.\n\nOverall, I think the current contributions of the paper aren't quite enough. The derivation of the fast LOOCV algorithm is straightforward linear algebra (I agree it's surprising that scikit-learn isn't implementing this method, but I think that's just more of a criticism of scikit learn). And the derivation of the EM algorithm iterates seems relatively straightforward from [26]. I think the real-data experiments are pretty good, but I'm not sure this is enough to carry the paper. It would be really interesting to investigate the proposed EM algorithm a bit more -- e.g. get into the motivation as above or study it theoretically (e.g. under what conditions is the assumption on line 208 that the number of iterations is $o(n)$ correct?)\n\nI had a few other more contained comments:\n- I don't think $q$ was defined mathematically in the paper.\n- On line 90, some non-standard notation for inner-products is used that I don't think was defined (maybe $\\tilde x_i^T \\hat\\beta_{(i)}$ was meant?)\n- The authors criticize LOOCV by saying \"For the squared-error risk to be minimax, and therefore donimate least-squares ... the number of grid points [in LOOCV] $L$ cannot be finite ...\". Does the proposed EM algorithm fix this problem? In particular, does it converge in a reasonable number of finite iterations, and, if not, is it still minimax if it is not run for infinite iterations?\n- In equation (8), what the expectations and covariances are with respect to should be specified -- I don't think any data-generating distribution was specified in the paper.\n- Line 147 \"we provide the derivations of (8) in the Appendix\" -- it should be specified where exactly in the appendix this can be found. This comment is true of every reference to the appendix throughout the paper -- readers shouldn't have to hunt through the whole thing to learn more about a specific point!\n- Line 174: is $s_j$ the same as $\\Sigma_{jj}$? If so, notation should be standardized. If not, I didn't see where $s_j$ was defined.\n-In equation (12), it should be clarified that the $\\tau^2 (p - n)_+$ is outside the summation.\n- Figure 2 is about showing that LOOCV is much slower than the EM algorithm. But the actual difference in runtime is like half a second at its maximum. I think it would be much more compelling to show a difference in runtime that is computationally meaningful.\n- Line 218: \"find the smallest value of $\\lambda_{max}$ that sets all the regression coefficient vector to zero\" -- isn't this typically $\\lambda_{max} = \\infty$ for ridge regression? \n\nI have a few other\n\nQUESTIONS:\nOverall, I think I'm feeling like the current paper has enough novelty in it for publication. But if the authors could argue against this, that'd be the most likely way to change my score -- I'm also curious what the other reviewers think on this issue.\n\nLIMITATIONS:\nI don't think limitations of the proposed method have been addressed in that they haven't really been studied -- it would be surprising for the proposed EM method to have no failure modes at all, but none are studied in the paper. \n\nI don't think there is any potential negative societal impact of this work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a simplified implementation of the EM algorithm of the Bayesian Ridge regression for the hyperparameter estimation. The advantage of the proposed method is in its computational cost when searching the hyperparameter. The numerical experiments support their proposal in terms of computational time. \nThe essential idea is to use SVD and compute all necessary quantities at preprocessing, and to lighten the computational load of hyperparameter search in the main loop by using them.\n\nSTRENGTHS:\nOriginality & Significance: This kind of trick (completing as many of the necessary calculations as possible as preprocessing) is, in my opinion, done by many researchers actually, though it may not be mentioned much in papers or textbooks. Especially, using SVD in preprocessing is a well-known technique in the family of linear models. It may be new to point out that using EM via Bayesian is less computationally expensive than LOOCV with standard ridge regression in the order of $n$, but I do not think this is a special property of ridge regression and is not expected in more general settings. For these reasons, I consider the originality and significance to be low. \n\nQuality: All the numerical experiments seem to be certainly conducted.　Numerical codes are also distributed.　In this sense, the quality is high.\n\nClarity: Most of the contents are well written, but some important parameters characterizing the computational costs such as $k$ and $q$ are not well explained. For this reason, the clarity is at the normal level.\n\nWEAKNESSES:\n- The basic idea of the research is not surprising. \n\n- The Bayesian ridge regression is different from the ridge regression in the level of concept, although the estimators tend to behave similarly. In that sense, I am not sure if it is suitable to compare these two things (EM in Bayes and LOOCV in the standard ridge.).\n\nQUESTIONS:\n- What $k$ and $q$? Please explain about them when they first appear in the text.\n\nLIMITATIONS:\nThe authors address their research limitations well. I think there is no concern about the potential societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis article proposes a new method based on an Expectation-Maximisation framework to train a Bayesian formulation of ridge regression for penalised regression. It is compared with the standard cross-validation approach for training the regularisation parameter lambda, instead iteratively updating the value algorithmically. Novel data pre-processing via SVD is proposed that accelerates not only the EM algorithm but also the standard glmnet algorithm. Extensive comparisons are made between the proposed EM method and the now-popular CV approach and computational costs are considered.\n\nSTRENGTHS:\nI enjoyed reading this paper. The very widely used model specification of penalised regression is reformulated in a Bayesian framework and the algorithm approaches the problem in a principled manner. The advantages compared to the standard cv.glmnet or python equivalent are fairly convincing. The algorithmic steps are explained clearly and the results are impressively thorough (24 data sets!). The usability of the method appears to be high, with the practical hurdles for using this in comparison with glmnet or similar appear to be low: this seems like it could become competitive with the present standard for biomarker analysis if presented and promoted appropriately, although the same could also be said of many other methods that are now in widespread use presently.\n\nWEAKNESSES:\nSome of the phrasing of the results is quite unclear and possibly suspicious: “Up to one order of magnitude” - that can mean a factor of two or a factor of ten, depending on who you ask.  Also “up to” suggests that this is the maximum difference rather than a bulk estimate that might be more informative in general. Similarly, towards the end: “Up to 49 times faster” - I would be more convinced by a summary involving a characterisation of average differences rather than a maximum value, that might be an outlier or a particularly flattering data set. If that average needs to be characterised as a function of p or n or similar to be accurate then so much the better.\n\nSome result cells are empty in Table 2: it is OK for these to not be present, but it is not explained why, so explaining more clearly to avoid suspicion would be nice.\n\nIt seems that some of the inspiration for this method comes from relatively recent related work: “The EM procedure discussed in this paper is based on a recently published procedure for learning sparse linear regression models” - please be more clear about the specific contribution of this work vs previous work, i.e. horseshoe vs ridge or similar.\n\nSome of the language and explanation could be tidied up slightly. Some examples:\n\n- \"that minimizes (3.” - typo?\n- There is no real difference between “missing data” and a latent variable in a Bayesian framework. If you could explain more clearly what insight is meant to be provided then that would be nice: if there is a clever trick here then please spell it out for those of us who haven't seen it before.\n- “Described in 5”? “Described in Section 5” maybe?\n- I assume the third order models include linear/quadratic terms and similar, but it’s slightly unclear.\n\nQUESTIONS:\nWhat is stopping this from becoming the norm if it is superior to glmnet/scikit-learn? One thing that people like about glmnet (not necessarily a good thing in statistical terms) is that it spits out specific predictors with specific values that are easy to interpret. Does this offer an end product that is similarly easy to interpret without any real domain expertise in inference to get it work? If so, is it necessarily a good thing to let non-statisticians use this method without understanding what's happening under the hood?\n\nWhat is the behaviour in contrast to CV with potentially non-i.i.d. Data? Does the new method make an analogous assumption to LOOCV, or avoid it? Can this difference be identified empirically?\n\ncv.glmnet solves for lambda.min and lambda.1se, i.e. the lambda value that minimises the objective function vs. one that is inflated a little for a little extra sparsity. Which is being compared with in the results here? And is there an analogous decision to make with the EM method? \n\nAre people likely to object to using explicit priors in a Bayesian method vs regularisation for a frequentist method? Are you able to convince people of the use of why it’s acceptable to use the ones you have chosen generically for any data set? What is the sensitivity to pi(tau)?\n\nIf you were really doing things at scale and worried about computational costs, it would be fairly straightforward to distribute the fixed grid evaluations of LOOCV in parallel with minimal overhead, effectively removing the factor of L that disadvantages the method. To what extent will the Bayes-EM formulation take advantage of similar distributed computing if available, and to what extent does this undermine the performance advantage represented here?\n\nAre there specific instances in which the EM algorithm doesn’t converge quickly? Are there good diagnostics for identifying when this is happening? Figure 2 seems to show slightly concerning behaviour in terms of number of k needed for increasing p. Do you have some theoretical characterisation of this bump, possibly representing the degree of non-Gaussianity of the posterior? It would be nice to know a priori what to expect from the EM algorithm, so a slow convergence rate will not come as a surprise. Presumably in this domain the LOOCV will have the advantage in a performance-time trade-off here, which is not super clear from the results as presented.\n\nLOOCV basically picks more or less at random from a pool of highly correlated predictors, which is not a good thing. Does BayesEM exhibit the same behaviour? Does this Bayesian formulation open up the possibility of adequately modelling correlation of coefficients in sparse regression via modelling some correlated beta posteriors?\n\nHow robust is this to non-normally distributed betas? One advantage of LOOCV is that the regularisation is an inferential decision rather than a modelling decision, in contrast to BayesEM, where you are making positive belief statements, which may be empirically false. If they are false, how much does that actually matter?\n\nLIMITATIONS:\nNo immediate negative societal impact to be concerned about. It seems to be a reformulation of a method that is already in widespread use.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

comments: SUMMARY:
This paper studies the comparison of cross validation (CV) and  expectation maximization (EM) on tuning the regularization hyper-parameter $\lambda$.
The authors explained the running time of these two methods and showed that EM is faster than CV.
Then, the authors also provided some experimental results to support the conclusion that EM is faster than CV and demonstrate that they have comparable predictive performances.

STRENGTHS:
The proposed EM approach for tuning $\lambda$ is quite interesting.
I have not seen this approach before.

WEAKNESSES:
The result is mostly experimental.
Most of the theoretical analyses are on the running time.
The authors may want to provide some theoretical explanation on why these two methods end up having comparable predictive performances.

QUESTIONS:
See weaknesses

LIMITATIONS:
na
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-04

comments: SUMMARY:
This paper proposes a new method for estimating the regularization parameter in ridge regression models that is faster, than the widely used leave one out cross validation (LOOCV) technique. The new method is derived from a Bayesian view for estimating this regularization parameter which, through standard Expectation Maximization procedure, lends itself to a nice optimization problem with two variables. The efficiency comes from the approach taken to solve this optimization problem; the Authors are able to formulate the solution using exact closed form expressions, together with a pre-processing step -- SVD for the data matrix -- that is computed only once at the beginning of the procedure. The paper demonstrates the accuracy and efficiency of the proposed method using an extensive set of experiments on simulated
and real world datasets, and compares its performance against the fast LOOCV with a fixed grid for the ridge/regularization parameter, as well as with the grid based on data-dependent heuristic available in R and Python packages for glmnet.

STRENGTHS:
The contribution of this paper is clear and interesting, and the derivation for the procedure is straight forward to follow. The paper is nicely written, well organized, and the experiments are extensive and sufficient. I believe this will be a good addition to the literature.

WEAKNESSES:
The missing pieces that require discussion and elaboration from the Authors is when this proposed method will be worse than the fast LOOCV? Any assumptions on the data distribution? And how sensitive is this method to the choice of $a$ and $b$ in Eq. 9?

QUESTIONS:
No questions at this point.

LIMITATIONS:
See my remark on missing discussions and point that require further elaboration.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06