krls: Kernel-based Regularized Least Squares (KRLS)

In KRLS: Kernel-Based Regularized Least Squares

Kernel-based Regularized Least Squares (KRLS)

Description

Function implements Kernel-Based Regularized Least Squares (KRLS), a machine learning method described in Hainmueller and Hazlett (2014) that allows users to solve regression and classification problems without manual specification search and strong functional form assumptions. KRLS finds the best fitting function by minimizing a Tikhonov regularization problem with a squared loss, using Gaussian Kernels as radial basis functions. KRLS reduces misspecification bias since it learns the functional form from the data. Yet, it nevertheless allows for interpretability and inference in ways similar to ordinary regression models. In particular, KRLS provides closed-form estimates for the predicted values, variances, and the pointwise partial derivatives that characterize the marginal effects of each independent variable at each data point in the covariate space. The distribution of pointwise marginal effects can be used to examine effect heterogeneity and or interactions.

Usage

krls(X = NULL, y = NULL, whichkernel = "gaussian", lambda = NULL,
sigma = NULL, derivative = TRUE, binary= TRUE, vcov=TRUE,
print.level = 1,L=NULL,U=NULL,tol=NULL,eigtrunc=NULL,data=NULL,
approx = c("auto", "none", "nystrom"), nystrom_m = NULL,
landmarks = NULL, landmark_method = c("random", "kmeans"),
nystrom_eps = sqrt(.Machine$double.eps),
landmark_seed = NULL,
lambda_method = c("loo", "gcv"),
cat_columns = NULL)

Arguments

X	For the matrix interface (the original): an N by D numeric data matrix that contains the values of D predictor variables for i=1,\ldots,N observations. The matrix may not contain missing values or constants. Note that no intercept is required since the function operates on demeaned data and subtracting the mean of y is equivalent to including an (unpenalized) intercept into the model. For the formula interface (added in 1.2-0): a two-sided formula of the form y ~ x1 + x2 + ... with the response on the left-hand side and the covariates on the right. data must be supplied. The intercept column is dropped automatically. Both interfaces produce the same fit; the formula form is purely an ergonomic wrapper.
y	N by 1 data numeric matrix or vector that contains the values of the response variable for all observations. This vector may not contain missing values. Ignored when X is a formula (the response is taken from the formula).
data	For the formula interface: a data.frame containing the variables referenced in the formula. Required when X is a formula; ignored otherwise.
whichkernel	String vector that specifies which kernel should be used. Must be one of gaussian, linear, poly1, poly2, poly3, or poly4 (see details). Default is gaussian.
lambda	A positive scalar that specifies the \lambda parameter for the regularizer (see details). It governs the tradeoff between model fit and complexity. By default, this parameter is chosen by minimizing the sum of the squared leave-one-out errors.
sigma	A positive scalar that specifies the bandwidth of the Gaussian kernel (see gausskernel for details). Default changed in 1.7-0. When sigma = NULL, krls() now selects the bandwidth that maximizes the off-diagonal variance of the kernel matrix K (see b_maxvarK), matching the convention used by kbal::b_maxvarK and gpss. The pre-1.7 default was sigma = ncol(X_processed); pin that value explicitly to reproduce older fits. A one-line message() is emitted on the first call where the new default kicks in.
derivative	Logical that specifies whether pointwise partial derivatives should be computed. Currently, derivatives are only implemented for the Gaussian Kernel.
binary	Logical that specifies whether first-differences instead of pointwise partial derivatives should be computed for binary predictors. Ignored unless derivative=TRUE.
vcov	Logical that specifies whether variance-covariance matrix for the choice coefficients c and fitted values should be computed. Note that derivative=TRUE requires that vcov=TRUE for the exact path. Under approx = "nystrom", vcov=TRUE computes conditional approximate variances for the landmark coefficients and average derivatives without storing the full fitted-value covariance matrix.
print.level	Positive integer that determines the level of printing. Set to 0 for no printing and 2 for more printing.
L	Non-negative scalar that determines the lower bound of the search window for the leave-one-out optimization to find \lambda. Default is NULL which means that the lower bound is found by using an algorithm outlined in lambdasearch.
U	Positive scalar that determines the upper bound of the search window for the leave-one-out optimization to find \lambda. Default is NULL which means that the upper bound is found by using an algorithm outlined in lambdasearch.
tol	Positive scalar that determines the tolerance used in the optimization routine used to find \lambda. Default is NULL which means that convergence is achieved when the difference in the sum of squared leave-one-out errors between the i and the i+1 iteration is less than N * 10^-3.
eigtrunc	Positive scalar that determines how much eignvalues should be trunacted for finding the upper bound of the search window in the algorithm outlined in lambdasearch. If eigtrunc is set to 10^-6 this means that we keep only eigenvalues that are 10^-6 as large as the first. Default is eigtrunc=NULL which means no truncation is used.
approx	String selecting the approximation regime. "auto" (the default) uses the exact N by N kernel when the sample size does not exceed the would-be landmark count (nystrom_m, default min(500, N)), and switches to the Nystrom approximation otherwise. A one-line message is emitted when the switch happens, so the choice is visible at the call site. Conditions that are unsupported under Nystrom (non-Gaussian kernel, non-null eigtrunc, derivative = TRUE with vcov = FALSE) fall through to the exact path silently. "none" forces the exact path regardless of sample size; this is the historical KRLS behavior and gives identical fits to KRLS \le 1.5-2. "nystrom" forces the low-rank Nystrom approximation: an explicit m-dimensional feature map replaces the full kernel and the ridge problem is solved in m-space. Time becomes O(N m^2 + m^3) and memory O(N m), so KRLS is feasible at N well beyond the exact path's ~5000-row ceiling. When vcov = TRUE, coefficient, prediction, and average-derivative standard errors are computed as conditional approximate Nystrom standard errors – conditional on the selected landmarks, fixed \lambda, and the low-rank feature approximation. The full N by N fitted-value covariance matrix is not stored. Only the Gaussian kernel is supported.
nystrom_m	Integer number of landmark points to use when approx = "nystrom". Ignored otherwise. Default is min(500, N), which keeps every row as a landmark for small samples and caps at 500 for larger ones. Larger nystrom_m approaches exact KRLS at a quadratic cost; smaller trades fit quality for runtime/memory. The internal approximation is K \approx C\,W_{reg}^{-1}\,C^\top where C = K(X, Z), W = K(Z, Z), and W_{reg} is W with eigenvalues floored at nystrom_eps * max(eigenvalues of W).
landmarks	Optional landmark specification for approx = "nystrom". One of: NULL (auto-select nystrom_m landmarks via landmark_method); an integer vector of row indices into X; or an m by D numeric matrix of landmark coordinates in the original X units. If indices are supplied, the resolved standardized landmark matrix and the original index vector are both stored on the fit object; if a matrix is supplied, only the matrix is stored.
landmark_method	String selecting the auto-selection rule for landmarks when landmarks = NULL. "random" (the default) samples uniformly without replacement and is bit-reproducible under set.seed(). "kmeans" runs stats::kmeans() on the standardized predictors and uses cluster centers as landmarks – typically more robust on imbalanced predictor distributions, but only approximately reproducible across R versions because the algorithm's initialization is implementation-dependent.
nystrom_eps	Positive scalar controlling the relative-ridge stabilization of the landmark kernel W. Eigenvalues of W below nystrom_eps * max(eigenvalues of W) are floored at that threshold before the feature map is built; this stabilizes the W^{-1/2} operation when landmarks are near-collinear in feature space. Documented as numerical stabilization, not a modeling parameter. Default is sqrt(.Machine$double.eps).
landmark_seed	Optional integer seed used only for landmark selection when landmarks = NULL and the chosen landmark_method is stochastic (random sampling or k-means). If supplied, the global .Random.seed is saved before landmark selection and restored on exit, so the caller's downstream RNG state is unaffected. This makes Nystrom fits bit-reproducible without forcing the user to remember a set.seed() call in their own code and without the silent-seed-reset failure mode that can plague simulations. Default is NULL, in which case landmark selection draws from whichever RNG state the caller has in effect.
cat_columns	Optional categorical-column declaration (new in 1.7-0). One of: NULL (the default; no one-hot expansion, but a single warning is emitted if columns of X look categorical – factor / character / logical / numeric with few unique values – and were not declared); a character vector of column names in X; or an integer vector of column indices. Declared columns are one-hot encoded with all levels (no reference cell) and multiplied by sqrt(0.5), matching the convention used in kbal::one_hot and gpss. The factor of sqrt(0.5) compensates for the doubled squared-Euclidean distance from carrying both the "in level k" and "not in level k" indicators, so two observations differing in one category contribute 1 (not 2) to ||x_i - x_j||^2 – equivalent to a Hamming distance. Continuous columns are still standardized to sd = 1. There is no autodetection. Pass cat_columns = integer(0) (or character(0)) to acknowledge that you have looked at X and nothing is categorical; this silences the heuristic warning.
lambda_method	String selecting the objective minimized to choose \lambda when lambda is not supplied. "loo" (the default) uses the closed-form leave-one-out sum of squared errors – the historical KRLS criterion. "gcv" uses generalized cross-validation, RSS(\lambda) / (1 - \mathrm{tr}(S(\lambda))/n)^2, computed in closed form from the kernel eigendecomposition (exact path) or the cached SVD of \Phi (Nystrom path). Both methods are evaluated at the same per-iteration cost; GCV is often preferred when one or two observations have disproportionate leverage. Ignored when lambda is supplied directly.

Details

krls implements the Kernel-based Regularized Least Squares (KRLS) estimator as described in Hainmueller and Hazlett (2014). Please consult this reference for any details.

Kernel-based Regularized Least Squares (KRLS) arises as a Tikhonov minimization problem with a squared loss. Assume we have data of the from y_i,\,x_i where i indexes observations, y_i \in R is the outcome and x_i \in R^D is a D-dimensional vector of predictor values. Then KRLS searches over a space of functions H and chooses the best fitting function f according to the rule:

argmin_{f \in H} \sum_i^N (y_i - f(x_i))^2 + \lambda ||f||_{H^2}

where (y_i - f(x_i))^2 is a loss function that computes how ‘wrong’ the function is at each observation i and || f ||_{H^2} is the regularizer that measures the complexity of the function according to the L_2 norm ||f||^2 = \int f(x)^2 dx. \lambda is the scalar regularization parameter that governs the tradeoff between model fit and complexity. By default, \lambda is chosen by minimizing the sum of the squared leave-one-out errors, but it can also be specified by the user in the lambda argument to implement other approaches.

Under fairly general conditions, the function that minimizes the regularized loss within the hypothesis space established by the choice of a (positive semidefinite) kernel function k(x_i,x_j) is of the form

f(x_j)= \sum_i^N c_i k(x_i,x_j)

where the kernel function k(x_i,x_j) measures the distance between two observations x_i and x_j and c_i is the choice coefficient for each observation i. Let K be the N by N kernel matrix with all pairwise distances K_ij=k(x_i,x_j) and c be the N by 1 vector of choice coefficients for all observations then in matrix notation the space is y=Kc.

Accordingly, the krls function solves the following minimization problem

argmin_{f \in H} \sum_i^n (y - Kc)'(y-Kc)+ \lambda c'Kc

which is convex in c and solved by c=(K +\lambda I)^-1 y where I is the identity matrix. Note that this linear solution provides a flexible fitted response surface that typically reduces misspecification bias because it can learn a wide range of nonlinear and or nonadditive functions of the predictors.

If vcov=TRUE is specified, exact KRLS also computes the variance-covariance matrix for the choice coefficients c and fitted values y=Kc based on a variance estimator developed in Hainmueller and Hazlett (2014). Note that both exact matrices are N by N and therefore this results in increased memory and computing time. Under approx = "nystrom", vcov=TRUE computes conditional approximate variances in the m-landmark feature space and does not store the full fitted-value covariance matrix.

By default, krls uses the Gaussian Kernel (whichkernel = "gaussian") given by

k(x_i,x_j)=exp(\frac{-|| x_i - x_j ||^2}{\sigma^2})

where ||x_i - x_j|| is the Euclidean distance. The kernel bandwidth \sigma^2 is set to D, the number of dimensions, by default, but the user can also specify other values using the sigma argument to implement other approaches.

If derivative=TRUE is specified, krls also computes the pointwise partial derivatives of the fitted function wrt to each predictor using the estimators developed in Hainmueller and Hazlett (2014). These can be used to examine the marginal effects of each predictor and how the marginal effects vary across the covariate space. Average derivatives are also computed with variances when vcov=TRUE. Note that the derivative=TRUE option results in increased computing time and is only supported for the Gaussian kernel, i.e. when whichkernel = "gaussian". For the exact path, derivative=TRUE requires vcov=TRUE; under approx = "nystrom", derivative=TRUE, vcov=FALSE returns derivative point estimates without standard errors.

If binary=TRUE is also specified, the function will identify binary predictors and return first differences for these predictors instead of partial derivatives. First differences are computed going from the minimum to the maximum value of each binary predictor. Note that first differences are more appropriate to summarize the effects for binary predictors (see Hainmueller and Hazlett (2014) for details).

A few other kernels are also implemented, but derivatives are currently not supported for these: "linear": k(x_i,x_j)=x_i'x_j, "poly1", "poly2", "poly3", "poly4" are polynomial kernels based on k(x_i,x_j)=(x_i'x_j +1)^p where p is the order.

Value

A list object of class krls with the following elements:

K	N by N matrix of pairwise kernel distances between observations.
coeffs	N by 1 vector of choice coefficients c.
Le	scalar with sum of squared leave-one-out errors.
fitted	N by 1 vector of fitted values.
X	original N by D predictor data matrix.
y	original N by 1 matrix of values of the outcome variable.
sigma	scalar with value of bandwidth, \sigma^2, used for the Gaussian kernel.
lambda	scalar with value of regularization parameter, \lambda, used (user specified or based on leave-one-out cross-validation).
R2	scalar with value of R-squared
vcov.c	Variance covariance matrix for choice coefficients (N by N for exact KRLS; m by m conditional approximate covariance under approx = "nystrom"; NULL unless vcov=TRUE is specified).
vcov.fitted	N by N variance covariance matrix for fitted values under exact KRLS (NULL unless vcov=TRUE is specified, and always NULL under approx = "nystrom" to preserve the memory benefit).
derivatives	N by D matrix of pointwise partial derivatives based on the Gaussian kernel (NULL unless derivative=TRUE is specified. If binary=TRUE is specified, first differences are returned for binary predictors.
avgderivatives	1 by D matrix of average derivative based on the Gaussian kernel (NULL unless derivative=TRUE is specified. If binary=TRUE is specified, average first differences are returned for binary predictors.
var.avgderivatives	1 by D matrix of variances for average derivative based on gaussian kernel (NULL unless derivative=TRUE is specified. If binary=TRUE is specified, variances for average first differences are returned for binary predictors.
binaryindicator	1 by D matrix that indicates for each predictor if it is treated as binary or not (evaluates to FALSE unless binary=TRUE is specified and a predictor is recognized binary.

Note

The function requires the storage of a N by N kernel matrix and can therefore exceed the memory limits for very large datasets.

Setting derivative=FALSE and vcov=FALSE is useful to reduce computing time if pointwise partial derivatives and or variance covariance matrices are not needed.

Author(s)

Jens Hainmueller (Stanford) and Chad Hazlett (UCLA)

References

Jeremy Ferwerda, Jens Hainmueller, Chad J. Hazlett (2017). Kernel-Based Regularized Least Squares in R (KRLS) and Stata (krls). Journal of Statistical Software, 79(3), 1-26. doi:10.18637/jss.v079.i03

Hainmueller, J. and Hazlett, C. (2014). Kernel Regularized Least Squares: Reducing Misspecification Bias with a Flexible and Interpretable Machine Learning Approach. Political Analysis, 22(2)

Rifkin, R. 2002. Everything Old is New Again: A fresh look at historical approaches in machine learning. Thesis, MIT. September, 2002.

Evgeniou, T., Pontil, M., and Poggio, T. (2000). Regularization networks and support vector machines. Advances In Computational Mathematics, 13(1):1-50.

Schoelkopf, B., Herbrich, R. and Smola, A.J. (2001) A generalized representer theorem. In 14th Annual Conference on Computational Learning Theory, pages 416-426.

Kimeldorf, G.S. Wahba, G. 1971. Some results on Tchebycheffian spline functions. Journal of Mathematical Analysis and Applications, 33:82-95.

See Also

predict.krls for fitted values and predictions. summary.krls for summary of the fit. plot.krls for plots of the fit.

Examples

# Linear example
# set up data
N <- 200
x1 <- rnorm(N)
x2 <- rbinom(N,size=1,prob=.2)
y <- x1 + .5*x2 + rnorm(N,0,.15)
X <- cbind(x1,x2)
# fit model
krlsout <- krls(X=X,y=y)
# summarize marginal effects and contribution of each variable
summary(krlsout)
# plot marginal effects and conditional expectation plots
plot(krlsout)


# non-linear example
# set up data
N <- 200
x1 <- rnorm(N)
x2 <- rbinom(N,size=1,prob=.2)
y <- x1^3 + .5*x2 + rnorm(N,0,.15)
X <- cbind(x1,x2)

# fit model
krlsout <- krls(X=X,y=y)
# summarize marginal effects and contribution of each variable
summary(krlsout)
# plot marginal effects and conditional expectation plots
plot(krlsout)

## 2D example:
# predictor data
X <- matrix(seq(-3,3,.1))
# true function
Ytrue <- sin(X)
# add noise 
Y     <- sin(X) + rnorm(length(X),sd=.3)
# approximate function using KRLS
out <- krls(y=Y,X=X)
# get fitted values and ses
fit <- predict(out,newdata=X,se.fit=TRUE)
# results
par(mfrow=c(2,1))
plot(y=Ytrue,x=X,type="l",col="red",ylim=c(-1.2,1.2),lwd=2,main="f(x)")
points(y=fit$fit,X,col="blue",pch=19)
arrows(y1=fit$fit+1.96*fit$se.fit,
       y0=fit$fit-1.96*fit$se.fit,
       x1=X,x0=X,col="blue",length=0)
legend("bottomright",legend=c("true f(x)=sin(x)","KRLS fitted f(x)"),
       lty=c(1,NA),pch=c(NA,19),lwd=c(2,NA),col=c("red","blue"),cex=.8)

plot(y=cos(X),x=X,type="l",col="red",ylim=c(-1.2,1.2),lwd=2,main="df(x)/dx")
points(y=out$derivatives,X,col="blue",pch=19)

legend("bottomright",legend=c("true df(x)/dx=cos(x)","KRLS fitted df(x)/dx"),
       lty=c(1,NA),pch=c(NA,19),lwd=c(2,NA),col=c("red","blue"),,cex=.8)

## 3D example
# plot true function
par(mfrow=c(1,2))
f<-function(x1,x2){ sin(x1)*cos(x2)}
x1 <- x2 <-seq(0,2*pi,.2)
z   <-outer(x1,x2,f)
persp(x1, x2, z,theta=30,main="true f(x1,x2)=sin(x1)cos(x2)")
# approximate function with KRLS
# data and outcomes
X <- cbind(sample(x1,200,replace=TRUE),sample(x2,200,replace=TRUE))
y   <- f(X[,1],X[,2])+ runif(nrow(X))
# fit surface
krlsout <- krls(X=X,y=y)
# plot fitted surface
ff  <- function(x1i,x2i,krlsout){predict(object=krlsout,newdata=cbind(x1i,x2i))$fit}
z   <- outer(x1,x2,ff,krlsout=krlsout)
persp(x1, x2, z,theta=30,main="KRLS fitted f(x1,x2)")

KRLS documentation built on June 5, 2026, 9:06 a.m.