# Introduction to Group Lasso and Elastic Net

The adelie package provides extremely efficient procedures for fitting the entire group lasso and group elastic net regularization path for GLMs, multinomial, the Cox model and multi-task Gaussian models. adelie is similar to the R package glmnet in scope of models, and in computational speed. The R package adelie is built from the same C++ code as used in the corresponding adelie Python package.

In this notebook, we give a brief overview of the group elastic net problem that adelie solves.

## Single-Response Group Elastic Net

The single-response group elastic net problem is given by \begin{align*} \mathrm{minimize}_{\beta, \beta_0} \quad& \ell(\eta) + \lambda \sum\limits_{g=1}^G \omega_g \left( \alpha \|\beta_g\|_2 + \frac{1-\alpha}{2} \|\beta_g\|_2^2 \right) \\\text{subject to}\quad& \eta = X \beta + \beta_0 \mathbf{1} + \eta^0 \end{align*} where $$\beta_0$$ is the intercept, $$\beta$$ is the coefficient vector, $$X$$ is the feature matrix, $$\eta^0$$ is a fixed offset vector, $$\lambda \geq 0$$ is the regularization parameter, $$G$$ is the number of groups, $$\omega_g \geq 0$$ is the penalty factor per group, $$\alpha \in [0,1]$$ is the elastic net parameter, and $$\beta_g$$ are the coefficients for the $$g$$ th group. $$\ell(\cdot)$$ is the loss function defined by the GLM. As an example, the Gaussian GLM (glm.gaussian()) defines the loss function as \begin{align*} \ell(\eta) &= \sum\limits_{i=1}^n w_i \left( -y_i \eta_i + \frac{\eta_i^2}{2} \right) \end{align*} where $$w \geq 0$$ is the observation weight vector, $$y$$ is the response vector, and $$\eta$$ is the linear prediction vector as in the optimization problem above. This is equivalent to the weighted sum-of-squared errors \begin{align*} \mbox{RSS} &= \frac12\sum\limits_{i=1}^n w_i \left( y_i -\eta_i \right)^2, \end{align*} since the term in $$\sum w_iy_i^2$$ is a constant for the optimization.

Specifically for the Gaussian GLM, we employ a specialized optimizer based on coordinate descent to solve the group elastic net problem.

The Gaussian GLM is written in “exponential family” form, with $$\eta_i$$ the natural parameter. Other GLMs such as binomial (glm.binomial()) and Poisson (glm.poisson()) have similar expressions for the negative log-likelihood. For other general GLMs as well as the Cox model (glm.cox()), we use a proximal Newton method, which leads to an Iterative Reweighted Least Squares (IRLS) algorithm, That is, we iteratively perform a quadratic approximation to $$\ell(\cdot)$$, which yields a sequence of Gaussian GLM group elastic net problems that we solve using our special solver based on coordinate descent.

The Gaussian GLM also admits a different algorithm, which we call the the covariance method, using summary statistics rather than individual-level data. The covariance method solves the following problem: \begin{align*} \mathrm{minimize}_{\beta} \quad& \frac{1}{2} \beta^\top A \beta - v^\top \beta + \lambda \sum\limits_{g=1}^G \omega_g \left( \alpha \|\beta_g\|_2 + \frac{1-\alpha}{2} \|\beta_g\|_2^2 \right) \end{align*} This method would be equivalent to the usual single-response Gaussian group elastic net problem if $$A \equiv X_c^\top W X_c$$ and $$v \equiv X_c^\top W y_c$$ where $$X_c$$ is column-centered version of $$X$$ and $$y_c$$ is the centered version of $$y-\eta^0$$ where the means are computed with weights $$W$$ (if intercept is to be fit).

This method only works for the Gaussian case since the proximal Newton method changes the weights $$W$$ at every IRLS iteration, so that without access to $$X$$, it is not possible to compute the new “$$A$$” and “$$v$$”.

## Multi-Response Group Elastic Net

The multi-response group elastic net problem is given by \begin{align*} \mathrm{minimize}_{B,\; \beta_0} \quad& \ell(\eta) + \lambda \sum\limits_{g=1}^G \omega_g \left( \alpha \|B_g\|_F + \frac{1-\alpha}{2} \|B_g\|_F^2 \right) \\\text{subject to}\quad& \eta = X B + \mathbf{1} \beta_0^\top + \eta^0 \end{align*} The model arises in “multitask” learning — glm.multigaussian() — as well as multinomial (multiclass) logistic regression — glm.multinomial(). This way, we have possibly different linear predictions for each of $$K$$ responses, and hence $$\eta$$ is $$n\times K$$. The coefficient “matrices” for each group $$B_g$$ are penalized using a Frobenius norm. Note that if an intercept is included in the model, an intercept is added for each response.

We use an alternative but equivalent representation in the adelie software, by “flattening” the coefficient matrices. Specifically we solve \begin{align*} \mathrm{minimize}_{\beta,\; \beta_0} \quad& \ell(\eta) + \lambda \sum\limits_{g=1}^G \omega_g \left( \alpha \|\beta_g\|_2 + \frac{1-\alpha}{2} \|\beta_g\|_2^2 \right) \\\text{subject to}\quad& \mathrm{vec}(\eta^\top) = (X \otimes I_K) \beta + (\mathbf{1} \otimes I_K) \beta_0 + \mathrm{vec}(\eta^{0\top}) \end{align*} where $$\mathrm{vec}(\cdot)$$ is the operator that flattens a column-major matrix into a vector, and $$A \otimes B$$ is the Kronecker product operator. Hence $$\beta \equiv \mathrm{vec}(B^\top)$$.

As indicated above, the multi-response group elastic net problem is technically of the same form as the single-response group elastic net problem. In fact, adelie reuses the single-response solver for multi-response problems by modifying the inputs appropriately (e.g. using matrix.kronecker_eye()) to represent $$X \otimes I_K$$). For the MultiGaussian family, we wrap the specialized single-response Gaussian solver and otherwise for general multi-response GLMs, we wrap the single-response GLM solver.

# Quickstart

In this section, we cover the basic usage of adelie.

library(adelie)

## Gaussian Group Elastic Net

Before using adelie, we assume that the user has a feature matrix X and a response vector y. For simplicity, we assume for now that X is a dense matrix, although we will later see that X can be substituted with a custom matrix as well. For demonstration, we will randomly generate our data.

n = 100     # number of observations
p = 1000    # number of features
set.seed(5) # seed
X = matrix(rnorm(n*p),n,p)
y = X[,1:10]%*%rnorm(10) + rnorm(n)*sqrt(10) # makes SNR = 1

### Lasso

The most basic call to grpnet simply supplies a X matrix and a glm object. For example, glm.gaussian() returns a Gaussian glm object, which corresponds to the usual least squares loss function. By default, grpnet will fit lasso by setting each feature as its own group. adelie is written to treat groups of size 1 in a more optimized manner, so it is a competitive lasso solver that has computation times similar to those of the glmnet package. Like glmnet, grpnet will also generate a sequence of 100 regularizations $$\lambda$$ evenly spaced on the log scale, by default if the user does not provide the path. Since grpnet is a path-solver, it will warm-start at the next $$\lambda$$ using the current solution on the path. For this reason, we recommend users to supply a sufficiently fine grid of $$\lambda$$ or use the default path!

fit = grpnet(X=X, glm=glm.gaussian(y=y))
print(fit)
##
## Call:  grpnet(X = X, glm = glm.gaussian(y = y))
##
##    Df  %Dev Lambda
## 1   0  0.00 3.3330
## 2   1  4.41 3.1810
## 3   1  8.43 3.0370
## 4   1 12.09 2.8990
## 5   1 15.42 2.7670
## 6   1 18.46 2.6410
## 7   1 21.23 2.5210
## 8   1 23.75 2.4070
## 9   1 26.05 2.2970
## 10  1 28.15 2.1930
## 11  1 30.06 2.0930
## 12  1 31.79 1.9980
## 13  1 33.38 1.9070
## 14  1 34.82 1.8210
## 15  1 36.14 1.7380
## 16  1 37.34 1.6590
## 17  2 38.73 1.5830
## 18  2 40.35 1.5110
## 19  2 41.83 1.4430
## 20  2 43.17 1.3770
## 21  2 44.40 1.3150
## 22  2 45.52 1.2550
## 23  3 46.55 1.1980
## 24  3 47.79 1.1430
## 25  3 48.91 1.0910
## 26  4 50.07 1.0420
## 27  5 51.52 0.9944
## 28  5 52.94 0.9492
## 29  6 54.43 0.9061
## 30  7 55.94 0.8649
## 31  9 57.44 0.8256
## 32  9 58.97 0.7881
## 33 10 60.41 0.7523
## 34 11 61.91 0.7181
## 35 16 63.65 0.6854
## 36 18 65.60 0.6543
## 37 19 67.43 0.6245
## 38 22 69.12 0.5962
## 39 23 70.82 0.5691
## 40 28 72.47 0.5432
## 41 30 74.05 0.5185
## 42 30 75.54 0.4949
## 43 32 76.98 0.4724
## 44 35 78.35 0.4510
## 45 38 79.66 0.4305
## 46 42 80.95 0.4109
## 47 43 82.18 0.3922
## 48 46 83.37 0.3744
## 49 50 84.55 0.3574
## 50 55 85.70 0.3411
## 51 58 86.79 0.3256
## 52 58 87.80 0.3108
## 53 60 88.72 0.2967
## 54 62 89.60 0.2832
## 55 65 90.43 0.2703

The print() methods gives a nice summary of the fit. Note that the solver finished early in this example. By default, the solver terminates if the latter reaches $$90\%$$ as a simple heuristic to avoid overfitting. This threshold can be controlled via the argument adev_tol.

The output of grpnet is a an object of class “grpnet”, which contains some simple descriptors of the model, as well as a state object that represents the state of the optimizer. For most use-cases, the users do not need to inspect the internals of a state object, but rather can extract the useful information via the methods print(), plot(), predict() and coef().

plot(fit)

Here we plot the coefficients profiles as a function of $$-\log(\lambda)$$. By default grpnet standardizes the features internally, and these coefficients are on the scale of the standardized features. One can avoid standardization via standardize = FALSE in the call to grpnet().

We can make predictions at new values for the features — here we use a subset of the original:

pred = predict(fit, newx = X[1:5,],lambda = c(1.5, 1))
pred
##             [,1]        [,2]
## [1,]  1.67467724  2.35995807
## [2,]  1.23844475  1.82265012
## [3,]  2.00210773  1.70850627
## [4,] -3.47439324 -4.30161563
## [5,] -0.06120398 -0.06572013

Finally, we would like to choose a good value for $$\lambda$$, and for that we use cross-validation. We include a progress_bar argument, which can be helful for big problems. The plot method for a cv.grpnet object displays the average cross-validated deviance of the model, with approximate standard error bars for the average.

fitcv = cv.grpnet(X,glm.gaussian(y),progress_bar = TRUE)
plot(fitcv)

Two vertical dashed lines are included: one at the value of $$\lambda$$ corresponding to the minimum value, and another at a larger (more conservative) value of $$\lambda$$, such that the mean error is within one standard error of the minimum.

One can print the fitted cv.glmnet object:

fitcv
##
## Call:  cv.grpnet(X = X, glm = glm.gaussian(y), progress_bar = TRUE)
##
## Measure: Mean Deviance
##
##     Lambda Index Measure    SE Nonzero
## min 0.7881    32   11.66 2.049       9
## 1se 1.3146    21   13.47 2.396       2

One can also predict directly from it:

pred = predict(fitcv, newx = X)
pred = predict(fitcv, newx = X, lambda = "lambda.min")

In the first case the prediction is at the default value of $$\lambda$$, which is lambda = "lambda.1se". Alternatively, any numeric values of $$\lambda$$ can be supplied.

### Group Lasso

To fit group lasso, the user simply needs to supply a groups argument that defines the starting column index of $$X$$ for each group. For example, if there are 4 features with two (contiguous) groups of sizes 3 and 1, respectively, then groups = c(1, 4). For demonstration, we take the same data as before and group every 10 features. We then fit group lasso using the same function.

fitg = grpnet(
X=X,
glm=glm.gaussian(y=y),
groups=seq(from = 1, to = p, by=10),
)
print(fitg)
##
## Call:  grpnet(X = X, glm = glm.gaussian(y = y), groups = seq(from = 1,      to = p, by = 10))
##
##     Df  %Dev Lambda
## 1    0  0.00 1.3680
## 2   10  5.72 1.3060
## 3   10 10.94 1.2470
## 4   10 15.71 1.1900
## 5   10 20.06 1.1360
## 6   10 24.03 1.0840
## 7   10 27.66 1.0350
## 8   10 30.97 0.9881
## 9   10 33.99 0.9432
## 10  10 36.75 0.9004
## 11  10 39.27 0.8594
## 12  10 41.57 0.8204
## 13  10 43.67 0.7831
## 14  10 45.58 0.7475
## 15  10 47.33 0.7135
## 16  10 48.93 0.6811
## 17  10 50.39 0.6501
## 18  10 51.72 0.6206
## 19  10 52.93 0.5924
## 20  10 54.04 0.5654
## 21  10 55.05 0.5397
## 22  10 55.98 0.5152
## 23  10 56.82 0.4918
## 24  20 58.01 0.4694
## 25  30 59.67 0.4481
## 26  30 61.38 0.4277
## 27  30 62.94 0.4083
## 28  30 64.38 0.3897
## 29  50 65.82 0.3720
## 30  50 67.37 0.3551
## 31 110 69.27 0.3390
## 32 130 71.36 0.3236
## 33 130 73.36 0.3089
## 34 140 75.24 0.2948
## 35 140 77.00 0.2814
## 36 140 78.62 0.2686
## 37 150 80.13 0.2564
## 38 170 81.56 0.2448
## 39 170 82.90 0.2336
## 40 180 84.16 0.2230
## 41 190 85.33 0.2129
## 42 200 86.40 0.2032
## 43 210 87.41 0.1940
## 44 220 88.37 0.1852
## 45 220 89.26 0.1767
## 46 230 90.10 0.1687
plot(fitg)

Notice in the printout the active set (df) increases in steps of 10, as expected. Also the coefficient plot colors all coefficients for a group with the sample color.

As before, we can use cross-validation to select the tuning parameter.

fitgcv = cv.grpnet(
X,
glm.gaussian(y),
groups=seq(from = 1, to = p, by=10),
progress_bar = TRUE)
plot(fitgcv)

## GLM Group Elastic Net

In the previous section, we covered how to solve penalized least squares regression. Nearly all of the content remains the same for fitting penalized GLM regression. The only difference is in the choice of the glm object. For brevity, we only discuss logistic regression as our non-trivial GLM example, however the following discussion applies for any GLM.

Let’s modify our example and generate a binomial response.

eta = X[,1:5]%*%rnorm(5)/sqrt(5)
mu = 1 / (1 + exp(-eta))
y = rbinom(n, size = 1, prob = mu)

To solve the group elastic net problem using the logistic loss, we simply provide the binomial GLM object. For simplicity, we fit the lasso problem below.

fitb = grpnet(X, glm.binomial(y))
plot(fitb)

Cross-validation works as before:

fitbcv = cv.grpnet(X,glm.binomial(y),progress_bar = TRUE)
plot(fitbcv)

We can make predictions from the fitted objects as before. For GLMs, the predictions are for the link ($$\eta$$).

We can specify coefficient groups just as before.

## Multi-Response GLM Elastic Net

grpnet is also able to fit multi-response GLM elastic nets. Currently, we only support MultiGaussian (least squares) and Multinomial GLMs.

The following code shows an example of fitting a multinomial regression problem. We will use the glm.multinomial() family, which expects a matrix response $$Y$$ with $$K$$ columns (the number of classes). Each row of $$Y$$ is a proportion vector which sums to 1. In the example below, $$Y$$ is an indicator matrix, with a single 1 in each row, the rest being zeros.

n = 300     # number of observations
p = 100    # number of features
K = 4       # number of classes
set.seed(7)
X = matrix(rnorm(n*p),n,p)
eta = X[, 1:5] %*% matrix(rnorm(K*5),5,K)/sqrt(5)
probs = exp(eta)
probs = probs/rowSums(probs)
Y = t(apply(probs,1,function(prob)rmultinom(1, 1, prob)))
Y[1:5,]
##      [,1] [,2] [,3] [,4]
## [1,]    0    1    0    0
## [2,]    0    0    0    1
## [3,]    0    1    0    0
## [4,]    0    0    1    0
## [5,]    1    0    0    0

The fitting proceeds as before. We will directly fit a group lasso, with groups of 5 chosen similarly to before.

grps = seq(from=1, to=p, by = 5)
fitm = grpnet(X, glm.multinomial(Y), groups=grps)
print(fitm)
##
## Call:  grpnet(X = X, glm = glm.multinomial(Y), groups = grps)
##
##     Df  %Dev    Lambda
## 1    0  0.00 0.0233300
## 2    5  2.02 0.0222700
## 3    5  3.94 0.0212600
## 4    5  5.70 0.0202900
## 5    5  7.33 0.0193700
## 6    5  8.83 0.0184900
## 7    5 10.22 0.0176500
## 8    5 11.50 0.0168400
## 9    5 12.70 0.0160800
## 10   5 13.81 0.0153500
## 11   5 14.84 0.0146500
## 12   5 15.80 0.0139800
## 13   5 16.69 0.0133500
## 14   5 17.53 0.0127400
## 15   5 18.30 0.0121600
## 16   5 19.03 0.0116100
## 17   5 19.71 0.0110800
## 18   5 20.35 0.0105800
## 19   5 20.94 0.0101000
## 20   5 21.49 0.0096390
## 21   5 22.01 0.0092010
## 22   5 22.50 0.0087830
## 23   5 22.95 0.0083840
## 24   5 23.37 0.0080020
## 25  10 24.00 0.0076390
## 26  15 24.66 0.0072920
## 27  15 25.50 0.0069600
## 28  15 26.23 0.0066440
## 29  20 27.00 0.0063420
## 30  30 27.94 0.0060540
## 31  35 29.01 0.0057780
## 32  35 30.12 0.0055160
## 33  50 31.29 0.0052650
## 34  55 32.61 0.0050260
## 35  65 33.97 0.0047970
## 36  65 35.40 0.0045790
## 37  75 36.81 0.0043710
## 38  75 38.26 0.0041730
## 39  85 39.73 0.0039830
## 40  90 41.20 0.0038020
## 41  95 42.67 0.0036290
## 42  95 44.09 0.0034640
## 43  95 45.47 0.0033070
## 44 100 46.81 0.0031560
## 45 100 48.12 0.0030130
## 46 100 49.40 0.0028760
## 47 100 50.64 0.0027450
## 48 100 51.84 0.0026200
## 49 100 53.01 0.0025010
## 50 100 54.15 0.0023880
## 51 100 55.26 0.0022790
## 52 100 56.35 0.0021760
## 53 100 57.41 0.0020770
## 54 100 58.44 0.0019820
## 55 100 59.45 0.0018920
## 56 100 60.44 0.0018060
## 57 100 61.40 0.0017240
## 58 100 62.34 0.0016460
## 59 100 63.27 0.0015710
## 60 100 64.17 0.0015000
## 61 100 65.06 0.0014310
## 62 100 65.93 0.0013660
## 63 100 66.78 0.0013040
## 64 100 67.62 0.0012450
## 65 100 68.44 0.0011880
## 66 100 69.25 0.0011340
## 67 100 70.04 0.0010830
## 68 100 70.82 0.0010340
## 69 100 71.59 0.0009866
## 70 100 72.35 0.0009417
## 71 100 73.09 0.0008989
## 72 100 73.83 0.0008581
## 73 100 74.56 0.0008191
## 74 100 75.28 0.0007819
## 75 100 75.99 0.0007463
## 76 100 76.69 0.0007124
## 77 100 77.39 0.0006800
## 78 100 78.07 0.0006491
## 79 100 78.76 0.0006196
## 80 100 79.43 0.0005914
## 81 100 80.10 0.0005646
## 82 100 80.76 0.0005389
## 83 100 81.41 0.0005144
## 84 100 82.05 0.0004910
## 85 100 82.69 0.0004687
## 86 100 83.31 0.0004474
## 87 100 83.72 0.0004271
## 88 100 84.44 0.0004077
## 89 100 85.08 0.0003891
## 90 100 85.47 0.0003714
## 91 100 86.16 0.0003546
## 92 100 86.76 0.0003384
## 93 100 87.13 0.0003231
## 94 100 87.78 0.0003084
## 95 100 88.14 0.0002944
## 96 100 88.76 0.0002810
## 97 100 89.11 0.0002682
## 98 100 89.70 0.0002560
## 99 100 90.02 0.0002444
plot(fitm)

The coefficient for each feature is a vector(one per class), so rather than show multiple coefficients, the plot shows the progress of the 2norm as a function of $$\lambda$$. Once again, because of the grouping (into groups of 5 here), the groups are colored the same. In this case, the first group has all the action, so appears much earlier than the rest.

fitmcv = cv.grpnet(X,glm.multinomial(Y),groups=grps)
plot(fitmcv)

Another important difference from the single-response case is that the user must be aware of the shape of the returned coefficients, as returned by coef(fitm) or coef(fitmcv).

names(coef(fitm))
## [1] "intercepts" "betas"      "df"         "lambda"

We see that coef() returns a list. For multi-response GLMs, the intercepts component is included by default for each response, and hence is a $$L \times K$$ matrix, where $$L$$ is the number of $$\lambda$$s in the path. The betas component will be a $$L \times pK)$$ sparse matrix where every successive $$K$$ columns correspond to the coefficients associated with a feature and across all responses.

Fortunately the predict() method understands this structure, and behaves as intended.

predict(fitmcv,newx = X[1:3,])
## , , 1
##
##            [,1]       [,2]       [,3]       [,4]
## [1,] -0.8315155  0.7267649  0.6423057 -1.0228404
## [2,]  0.4868225 -0.5441420 -0.6714108  0.2398791
## [3,] -0.1516871  0.3078583 -1.0663172  0.4250282

Here by default the prediction is made at lambda = "lambda.1se".

## Cox Models

Our final example will be for censored survival data, and we will fit Cox proportional hazards model.

We create an example with a $$500 \times 100$$ sparse $$X$$ matrix. In this example we have right censoring. So the “subject” exits at times in y, and the binary status is 1 of the exit is a “death”, else 0 if censored.

set.seed(9)
n <- 500
p <- 100
X  <- matrix(rnorm(n*p), n, p)
X[sample.int(n * p, size = 0.5 * n * p)] <- 0
X_sparse <- Matrix::Matrix(X, sparse = TRUE)

nzc <- p / 4
beta <- rnorm(nzc)
fx <- X[, seq(nzc)] %*% beta / 3
hx <- exp(fx)
y <- rexp(n,hx)
status <- rbinom(n = n, prob = 0.3, size = 1)

We now fit an elastic-net model using the glm.cox() family, with every set of 5 coefficients in a group.

groups = seq(from = 1, to = p, by = 5)
fitcv <- cv.grpnet(X_sparse,
glm.cox(stop = y, status = status),
alpha = 0.5,
groups = groups)
par(mfrow = c(1,2))
plot(fitcv)
plot(fitcv\$grpnet.fit)