Lecture 6: Variable Selection - LASSOlu/ST7901/lecture notes/2019Lect6_shrink... · Oracle Properties Oracle Properties Motivations Shrinkage method for variable selection a continuous

Lecture 6: Variable Selection - LASSO

Wenbin Lu

Department of StatisticsNorth Carolina State University

Fall 2019

Wenbin Lu (NCSU) Data Mining and Machine Learning Fall 2019 1 / 45

Outline

Modern Penalization Methods

Lq penalty, ridge

LASSO

computation, solution path, LARS algorithmsparameter tuning, cross validationtheoretical properties

ridge regression


Oracle Properties

Notations

data (Xi ,Yi ), i = 1, · · · , nn: sample size

d : the number of predictors, Xi ∈ Rd .the full index set S = {1, 2, · · · , d}.the selected index set given by a procedure is Â, its size is |Â|.the linear coefficient vector β = (β1, · · · , βd)T .the true linear coefficients β0 = (β10, · · · , βd0)T .the true model A0 = {j : j = 1, · · · , d , |βj0| 6= 0}.


Oracle Properties Oracle Properties

Motivations

Shrinkage method for variable selection a continuous process

more stable, not suffering from high variability.

one unified selection and estimation framework; easy to makeinferences

advances in theories: oracle properties, valid asymptotic inferences

In general, we standardize the inputs first before applying the methods.

X ’s are centered to mean 0 and variance 1

y is centered to mean 0

We often fit a model without an intercept.



Ideal Variable Selection

The best results we can expect from a selection procedure:

able to obtain the correct model structure

keep all the important variables in the modelfilter all the noisy variable out of the model

has some optimal inferential properties

consistent, optimal convergence rateasymptotic normality, most efficient

True model: yi = β00 +∑d

j=1 xijβj0 + �i ,

important index set: A0 = {j : βj0 6= 0, j = 1, · · · , d}unimportant index set: Ac0 = {j : βj0 = 0, j = 1, · · · , d}



Oracle Properties

An oracle performs as if the true model were known:

1 selection consistency

β̂j 6= 0 for j ∈ A0; β̂j = 0 for j ∈ Ac0;

2 estimation consistency:

√n(β̂A0 − βA0)→d N(0,ΣI ),

βA0 = {βj , j ∈ A0} and ΣI is the covariance matrix if knowing thetrue model.


Shrinkage Methods

Penalized Regularization Framework

One revolutionary work is the development of regularization framework:

minβ

L(β; y,X ) + λJ(β)

L(β; y,X ) is the loss function

For OLS, L =∑n

i=1[yi − β0 −∑d

j=1 βjxij ]2

For MLE methods, L = log likelihood

For Cox’s PH models, L is the partial likelihood

In supervised learning, L is the hinge loss function (SVM), orexponential loss (AdaBoost)


Shrinkage Methods

Lq Shrinkage Penalty

J(β) is the penalty function.

Jq(|β|) = λ‖β‖qq =d∑

j=1

|βj |q, q ≥ 0.

J0(|β|) =∑d

j=1 I (βj 6= 0) (L0; Donoho and Johnstone 1988)

J1(|β|) =∑d

j=1 |βj | (LASSO; Tibshirani 1996)

J2(|β|) =∑d

j=1 β2j (Ridge; Hoerl and Kennard, 1970)

J∞(|β|) = maxj |βj | (Supnorm penalty; Zhang et al. 2008)


Shrinkage Methods

Elements of Statistial Learning

Hastie, Tibshirani & Friedman 2001 Chapter 3

PSfrag replaementsq = 4 q = 2 q = 1 q = 0:5 q = 0:1

Figure 3.13: Contours of onstant value of Pj j�j jqfor given values of q.


Shrinkage Methods LASSO

The LASSO

Applying the absolute penalty on the coefficients

minβ

n∑i=1

(yi −d∑

j=1

xijβj)2 + λ

d∑j=1

|βj |

λ ≥ 0 is a tuning parameterλ controls the amount of shrinkage; the larger λ, the greater amountof shrinkage

What happens if λ→ 0? (no penalty)What happens if λ→∞?



Equivalent formulation for LASSO

minβ

n∑i=1

(yi −d∑

j=1

xijβj)2, subject to

d∑j=1

|βj | ≤ t.

Explicitly apply the magnitude constraint on the parameters

Making t sufficiently small will cause some coefficients to be exactlyzero.

The lasso is a kind of continuous subset selection





β^ β^2. .β

1

β2

β1 β

Figure 3.12: Estimation piture for the lasso (left)and ridge regression (right). Shown are ontours of theerror and onstraint funtions. The solid blue areas arethe onstraint regions j�1j+ j�2j � t and �21 + �22 � t2,respetively, while the red ellipses are the ontours ofthe least squares error funtion.



LASSO Solution: soft thresholding

For the orthogonal design case with X ′X = I , the LASSO method isequivalent to: solve βj ’s componentwisely by solving

minβj

(βj − β̂lsj )2 + λ|βj |

The solution to the above problem is

β̂lassoj = sign(β̂lsj )(|β̂lsj | −

λ

2)+

=

β̂lsj −

λ2 if β̂

lsj >

λ2

0 if |β̂lsj | ≤λ2

β̂lsj +λ2 if β̂

lsj < −

λ2

shrinks big coefficients by a constant λ2 towards zero.

truncates small coefficients to zero exactly.



Comparison: Different Methods Under Orthogonal Design

When X is orthonormal, i.e. XTX = I

Best subset (of size M): β̂lsj if rank(|β̂lsj |) ≤ M.keeps the largest coefficients

Ridge regression: β̂lsj /(1 + λ)

does a proportional shrinkage

Lasso: sign(β̂lsj )(|β̂lsj | −λ2 )+

transform each coefficient by a constant factor first, then truncate it atzero with a certain threshold“soft thresholding”, used often in wavelet-based smoothing


Shrinkage Methods Parameter Tuning of LASSO

How to Choose t or λ

The tuning parameter t or λ should be adaptively chosen to minimize theMSE or PE. Common tuning methods include

the tuning error, cross validation, leave-out-one cross validation(LOOCV), AIC, or BIC.

If t is chosen larger than t0 =∑d

j=1 |β̂lsj |, then β̂lassoj = β̂lsj .If t = t0/2, the least squares coefficients are shrunk by 50%.

In practice, we sometimes standardize t by s = t/t0 in [0, 1], and choosethe best value s from [0, 1].



K-fold Cross Validation

The simplest and most popular way to estimate the tuning error.1 Randomly split the data into K roughly equal parts2 For each k = 1, ...,K

leave the kth portion out, and fit the model using the other K − 1parts. Denote the solution by β̂−k

calculate prediction errors of β̂−k on the left-out kth portion

3 Average the prediction errors

Define the Index function (allocating memberships)

κ : {1, ..., n} → {1, ...,K}.

The K -fold cross-validation estimate of prediction error (PE) is

CV =1

n

n∑i=1

(yi − xTi β̂−κ(i)

)2Typical choice: K = 5, 10



Choose λ

Choose a sequence of λ values.

For each λ, fit the LASSO and denote the solution by β̂lassoλ .

Compute the CV (λ) curve as

CV (λ) =1

n

n∑i=1

(yi − xTi β̂

−κ(i)λ

),

which provides an estimate of the test error curve.

Find the best parameter λ∗ which minimizes CV (λ).

Fit the final LASSO model with λ∗. The final solution is denoted asβ̂lassoλ∗



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One-standard Error for CV

One-standard error rule:

choose the most parsimonious model with error no more than onestandard error above the best error.

used often in model selection (for a smaller model size).



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Shrinkage Methods Computation of LASSO

Computation of LASSO

Quadratic Programming

Shooting Algorithms (Fu 1998; Zhang and Lu, 2007)

LARS solution path (Efron et al. 2004)

the most efficient algorithm for LASSOdesigned for standard linear modelsR package ”lars”

Coordinate Descent Algorithm (CDA; Friedman et al. 2010)

designed for GLMsuitable for ultra-high dimensional problemsR package ”glmnet”



R package: lars

Provides the entire sequence of coefficients and fits, starting from zero, tothe least squares fit.

library(lars)

lars(x, y, type = c("lasso", "lar", "forward.stagewise",

"stepwise"), trace = FALSE, normalize = TRUE,

intercept = TRUE, Gram, eps = .Machine$double.eps,

max.steps, use.Gram = TRUE)

A “lars” object is returned, for which print, plot, predict, coef andsummary methods exist.

Details in Efron et al. (2004).

With ”lasso” option, it computes the complete lasso solutionsimultaneously for ALL values of the shrinkage parameter (λ or t ors) in the same computational cost as a least squares fit.



Details About LARS function

x : matrix of predictors

y : response

type: One of ”lasso”, ”lar”, ”forward.stagewise” or ”stepwise”, allvariants of Lasso. Default is ”lasso”.

trace: if TRUE, lars prints out its progress

normalize: if TRUE, each variable is standardized to have unit L2norm, otherwise it is left alone. Default is TRUE.

intercept: if TRUE, an intercept is included (not penalized),otherwise no intercept is included. Default is TRUE.

Gram: XTX matrix; useful for repeated runs (bootstrap) where alarge XTX stays the same.

eps: An effective zero

max.steps: the number of steps taken. Default is8 min(d , n − intercept), n is the sample size.Wenbin Lu (NCSU) Data Mining and Machine Learning Fall 2019 23 / 45




Shrinkage Factor s

Coeff

icients

0.0 0.2 0.4 0.6 0.8 1.0

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Figure 3.9: Pro�les of lasso oeÆients, as tuningparameter t is varied. CoeÆients are plotted versuss = t=Pp1 j�̂j j. A vertial line is drawn at s = 0:5, thevalue hosen by ross-validation. Compare Figure 3.7on page 7; the lasso pro�les hit zero, while those forridge do not.Wenbin Lu (NCSU) Data Mining and Machine Learning Fall 2019 24 / 45


Prediction for LARS

predict(object, newx, s, type = c("fit", "coefficients"),

mode = c("step", "fraction", "norm", "lambda"), ...)

coef(object, ...)

object: A fitted lars object

newx: If type=“fit”, then newx should be the x values at which thefit is required. If type=“coefficients”, then newx can be omitted.

s: a value, or vector of values, indexing the path. Its values dependson the mode . By default (mode=“step”), s takes on values between0 and p.

type: If type=“fit”, predict returns the fitted values. Iftype=”coefficients”, predict returns the coefficients.



“Mode” for Prediction in LARS

If mode=“step”, then s should be the argument indexes the lars stepnumber, and the coefficients will be returned corresponding to thevalues corresponding to step s.

If mode=“fraction”, then s should be a number between 0 and 1, andit refers to the ratio of the L1 norm of the coefficient vector, relativeto the norm at the full LS solution.

If mode=“norm”, then s refers to the L1 norm of the coefficientvector.

If mode= λ, then s is the lasso regularization parameter



Example: Diabetes Data

library(lars)

data(diabetes)

attach(diabetes)

## fit LASSO

object


R Code to Compute K-CV Error Curve for Lars

cv.lars(x, y, K = 10, index, type = c("lasso",

"lar", "forward.stagewise", "stepwise"),

mode=c("fraction", "step"), ...)

x : Input to lars

y : Input to lars

K : Number of folds

index: Abscissa values at which CV curve should be computed. Ifmode=”fraction”, this is the fraction of the saturated |β|. Thedefault value index=seq(from = 0, to = 1, length = 100). Ifmode=”step”, this is the number of steps in lars procedure.



R Code: CV.LARS

attach(diabetes)

lasso.s

Shrinkage Methods Consistency Properties of LASSO

Consistency - Definition

Let the true parameters be β0 and their estimates by β̂n. The subscript ‘n’means the sample size n.

Estimation consistency

β̂n − β0 →p 0, as n→∞.

Model selection consistency

P({j : β̂j 6= 0} = {j : β0j 6= 0}

)→ 1, as n→∞.

Sign consistency

P(β̂n =s β0

)→ 1, as n→∞,

whereβ̂n =s β0 ⇔ sign(β̂n) = sign(β0).

Sign consistency is stronger than model selection consistency.Wenbin Lu (NCSU) Data Mining and Machine Learning Fall 2019 30 / 45


Consistency of LASSO Estimator

Knight and Fu (2000) have shown that

Estimation Consistency: The LASSO solution is of estimationconsistency for fixed p. In other words,

β̂lasso(λn)→p β, as λn = o(n).

It is root-n consistent and asymptotically normal.

Model Selection Property: For λn ∝ n12 , as n→∞, there is a

non-vanishing positive probability for lasso to select the true model.

The l1 penalization approach is called basis pursuit in signal processing(Chen, Donoho, and Saunders, 2001).



Model Selection Consistency of LASSO Estimator

Zhao and Yu (2006). On Model Selection Consistency of Lasso. JMLR, 7,2541–2563.

If an irrelevant predictor is highly correlated with the predictors in thetrue model, Lasso may not be able to distinguish it from the truepredictors with any amount of data and any amount of regularization.

Under Irrepresentable Condition (IC), the LASSO is modelselection consistent in both fixed and large p settings.

The IC condition is represented as

|[Xn(1)TXn(1)]−1XTn (1)Xn(2)| < 1− η,

i.e., the regression coefficients of irrevelant covariates Xn(2) on therelevant covariates Xn(1) is constrained. It is almost necessary andsufficient for Lasso to select the true model.



Special Cases for LASSO to be Selection Consistent

Assume the true model size |A0| = q.The underlying model must satisfy a nontrivial condition if the lassovariable selection is consistent (Zou, 2006).

The lasso is always consistent in model selection under the followingspecial cases:

when p = 2;when the design is orthgonal;when the covariates have bounded constant correlation, 0 < rn <

11+cq

for some c ;when the design has power decay correlation with corr(Xi ,Xj) = ρ

|i−j|n ,

where |ρn| ≤ c < 1.


Ridge Penalty

Ridge Regression (q = 2)

Applying the squared penalty on the coefficients

minβ

n∑i=1

(yi −d∑

j=1

xijβj)2 + λ

d∑j=1

β2j

λ ≥ 0 is a tuning parameterThe solution

β̂ridge = (XTX + λId)−1X

Ty.

Note the similarity to the ordinary least squares solution, but with theaddition of a “ridge” down the diagonal.

As λ→ 0, β̂ridge → β̂ols.As λ→∞, β̂ridge → 0.


Ridge Penalty

Ridge Regression Solution

In the special case of an orthonormal design matrix,

β̂ridgej =

β̂olsj1 + λ

.

This illustrates the essential “shrinkage” feature of ridge regression

Applying the ridge regression penalty has the effect of shrinking theestimates toward zero

The penalty introducing bias but reducing the variance of the estimate

Ridge estimator does not threshold, since the shrinkage is smooth(proportional to the original coefficient).


Ridge Penalty

Invertibility

If the design matrix X is not full rank, then

XTX is not invertible,

For OLS estimator,, there is no unique solution for β̂ols.

This problem does not occur with ridge regression, however.

Theorem: For any design matrix X , the quantity XTX + λId is always

invertible; thus, there is always a unique solution β̂ridge.

For the ridge estimator, an unbiased estimate for its df is

d̂f (β̂ridge) = Tr{X (XTX + λId)−1XT}.


Ridge Penalty

Bias and Variance

The variance of the ridge regression estimate is

Var(β̂ridge) = σ2WXTXW ,

where W = (XTX + λId)−1.

The bias of the ridge regression estimate is

Bias(β̂ridge) = −λWβ.

It can be shown that

the total variance∑d

j=1 Var(β̂ridgej ) is a monotone decreasing

sequence with respect to λ

while the total squared bias∑d

j=1 Bias2(β̂

ridgej ) is a monotone

increasing sequence with respect to λ.


Ridge Penalty

Mean Squared Error (MSE) of Ridge

Existence Theorem:There always exists a λ such that the MSE of β̂ridge is less than the MSE

of β̂ols.

a rather surprising result with somewhat radical implications: even ifthe model we fit is exactly correct and follows the exact distributionwe specify, we can always obtain a better (in terms of MSE)estimator by shrinking towards zero.


Ridge Penalty

Bayesian Interpretation of Ridge Estimation

Penalized regression can be interpreted in a Bayesian context:

Suppose y = Xβ + �, where � ∼ N(0, σ2In).Given the prior β ∼ N(0, τ2Id), then the posterior mean of β giventhe data is

(XTX +σ2

τ2Id)−1X

Ty.


Ridge Penalty

R package: lm.ridge

library(MASS)

lm.ridge(formula, data, subset, na.action, lambda)

formula: a formula expression as for regression models, of form

response ∼ predictorsdata: an optional data frame in which to interpret the variablesoccurring in formula.

subset: expression saying which subset of the rows of the data shouldbe used in the fit. All observations are included by default.

na.action a function to filter missing data.

lambda: A scalar or vector of ridge constants.

If an intercept is present, its coefficient is not penalized.


Ridge Penalty

Output for lm.ridge

Returns a list with components

coef: matrix of coefficients, one row for each value of lambda. Notethat these are not on the original scale and are for use by the coefmethod.

scales: scalings used on the X matrix.

Inter: was intercept included?

lambda: vector of lambda values

ym: mean of y

xm: column means of x matrix

GCV: vector of GCV values


Ridge Penalty

Example for Ridge Regression

library(MASS)

longley

names(longley)[1]

Ridge Penalty


Hastie, Tibshirani & Friedman 2001 Chapter 3Co

efficie

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PSfrag replaements df(�)Figure 3.7: Pro�les of ridge oeÆients for theprostate aner example, as tuning parameter � is var-ied. CoeÆients are plotted versus df(�), the e�e-tive degrees of freedom. A vertial line is drawn atdf = 4:16, the value hosen by ross-validation.


Ridge Penalty Ridge vs LASSO

Lasso vs Ridge

Ridge regression:

is a continuous shrinkage method; achieves better predictionperformance than OLS through a biase-variance trade-off.

cannot produce a parsimonious model, for it always keeps all thepredictors in the model.

The lasso

does both continuous shrinkage and automatic variable selectionsimultaneously.


Ridge Penalty Ridge vs LASSO

Empirical Comparison

Tibshirani (1996) and Fu (1998) compared the prediction performance ofthe lasso, ridge and bridge regression (Frank and Friedman, 1993) andfound

None of them uniformly dominates the other two.

However, as variable selection becomes increasingly important inmodern data analysis, the lasso is much more appealing owing to itssparse representation.


Oracle PropertiesOracle Properties

Shrinkage MethodsLASSOParameter Tuning of LASSOComputation of LASSOConsistency Properties of LASSO

Ridge Penalty

Documents

Lecture 6: Variable Selection - LASSOlu/ST7901/lecture notes/2019Lect6_shrink... · Oracle Properties Oracle Properties Motivations Shrinkage method for variable selection a continuous