Solve a randomly perturbed LASSO problem.
randomizedLasso(X,
y,
lam,
family=c("gaussian", "binomial"),
noise_scale=NULL,
ridge_term=NULL,
max_iter=100,
kkt_tol=1.e-4,
parameter_tol=1.e-8,
objective_tol=1.e-8,
objective_stop=FALSE,
kkt_stop=TRUE,
parameter_stop=TRUE)Matrix of predictors (n by p);
Vector of outcomes (length n)
Value of lambda used to compute beta. See the above warning
Be careful! This function uses the "standard" lasso objective
$$
1/2 \|y - x \beta\|_2^2 + \lambda \|\beta\|_1.
$$
In contrast, glmnet multiplies the first term by a factor of 1/n.
So after running glmnet, to extract the beta corresponding to a value lambda,
you need to use beta = coef(obj, s=lambda/n)[-1],
where obj is the object returned by glmnet (and [-1] removes the intercept,
which glmnet always puts in the first component)
Response type: "gaussian" (default), "binomial".
Scale of Gaussian noise added to objective. Default is 0.5 * sd(y) times the sqrt of the mean of the trace of X^TX.
A small "elastic net" or ridge penalty is added to ensure the randomized problem has a solution. 0.5 * sd(y) times the sqrt of the mean of the trace of X^TX divided by sqrt(n).
How many rounds of updates used of coordinate descent in solving randomized LASSO.
Tolerance for checking convergence based on KKT conditions.
Tolerance for checking convergence based on convergence of parameters.
Tolerance for checking convergence based on convergence of objective value.
Should we use KKT check to determine when to stop?
Should we use convergence of parameters to determine when to stop?
Should we use convergence of objective value to determine when to stop?
Design matrix.
Response vector.
Vector of penalty parameters.
Family: "gaussian" or "binomial".
Set of non-zero coefficients in randomized solution that were penalized. Integers from 1:p.
Set of zero coefficients in randomized solution. Integers from 1:p.
Set of non-zero coefficients in randomized solution that were not penalized. Integers from 1:p.
The sign pattern of the randomized solution.
List describing sampling parameters for conditional law of all optimization variables given the data in the LASSO problem.
List describing sampling parameters for conditional law of only the scaling variables given the data and the observed subgradient in the LASSO problem.
Affine transformation describing relationship between internal representation of the data and the data compontent of score of the likelihood at the unregularized MLE based on the sign_vector (a.k.a. relaxed LASSO).
Data component of the score at the unregularized MLE.
SD of Gaussian noise used to draw the perturbed objective.
The randomized solution. Inference is made conditional on its sign vector (so no more snooping of this value is formally permitted.)
If condition_subgrad == TRUE when sampling, then we may snoop on the observed subgradient.
The random vector in the linear term added to the objective.
For family="gaussian" this function uses the "standard" lasso objective
$$
1/2 \|y - x \beta\|_2^2 + \lambda \|\beta\|_1
$$
and adds a term
$$
- \omega^T\beta + \frac{\epsilon}{2} \|\beta\|^2_2
$$
where omega is drawn from IID normals with standard deviation
noise_scale and epsilon given by ridge_term.
See below for default values of noise_scale and ridge_term.
For family="binomial", the squared error loss is replaced by the
negative of the logistic log-likelihood.
Xiaoying Tian, and Jonathan Taylor (2015). Selective inference with a randomized response. arxiv.org:1507.06739
Xiaoying Tian, Snigdha Panigrahi, Jelena Markovic, Nan Bi and Jonathan Taylor (2016). Selective inference after solving a convex problem. arxiv:1609.05609
# NOT RUN {
set.seed(43)
n = 50
p = 10
sigma = 0.2
lam = 0.5
X = matrix(rnorm(n*p), n, p)
X = scale(X, TRUE, TRUE) / sqrt(n-1)
beta = c(3,2,rep(0,p-2))
y = X%*%beta + sigma*rnorm(n)
result = randomizedLasso(X, y, lam)
# }
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