A simple explanation of the Lasso and Least Angle Regression
Give a set of input measurements x1, x2 ...xp and an
outcome measurement y, the lasso fits a linear model
yhat=b0 + b1*x1+ b2*x2 + ... bp*xp
The criterion it uses is:
Minimize sum( (y-yhat)^2 ) subject to sum[absolute value(bj)] <= s
The first sum is taken over observations (cases) in the dataset.
The bound "s" is a tuning parameter. When "s" is large enough,
the constraint
has no effect and the solution is just the usual multiple linear least
squares regression of y on x1, x2, ...xp.
However when for smaller values of s (s>=0) the solutions are shrunken versions of the least squares estimates. Often, some of the coefficients bj are
zero. Choosing "s" is like choosing the number of predictors to use in a regression
model, and cross-validation is a good tool for estimating the best value for "s".
Computation of the Lasso solutions
The computation of the lasso solutions is a quadratic programming problem, and can be
tackled by standard numerical analysis algorithms.
But the least angle regression procedure is a better approach.
This algorithm exploits the special
structure of the lasso problem, and provides an efficient way to
compute the solutions simulataneously for all values of "s".
Least angle regression is like a more "democratic" version of
forward stepwise regression. Recall how forward stepwise regression
works:
Forward stepwise regression algorithm:
- Start with all coefficients bj equal to zero.
- Find the predictor xj most correlated with y, and add it
into the model. Take residuals r= y-yhat.
- Continue, at each stage adding to the model the predictor most correlated with r.
- Until: all predictors are in the model
The least angle regression procedure follows the same general scheme,
but doesn't add a predictor fully into the model. The coefficient
of that predictor is increased only until that predictor is no
longer the one most correlated with the residual r.
Then some other competing predictor is invited to "join the club".
Least angle regression algorithm:
- Start with all coefficients bj equal to zero.
- Find the predictor xj most correlated with y
- Increase the coefficient bj in the direction of the sign
of its correlation with y. Take residuals r=y-yhat along the way.
Stop when some other predictor xk
has as much correlation with r as xj has.
- Increase (bj, bk) in their joint least squares direction,
until some other predictor xm has as much correlation with the residual r.
- Continue until: all predictors are in the model
Surprisingly it can be shown that, with one modification, this procedure gives the entire
path of lasso solutions, as s is varied from 0 to infinity.
The modification needed is: if a non-zero coefficient hits zero,
remove it from the active set of predictors and recompute the
joint direction.