Regression with Lasso, all coeffs are 0

Question

I am currently experimenting Lasso with scikit in the case of high dimension. The labels are Y_i (real numbers), and the feature are X_i (X_i is a vector of size d=112). I have only three couples of (Y_i,X_i).

d>>n=3 so we are in the high dimension case.

import numpy as np

Y = np.array([ 0.24186978,  0.20693342,  0.00441244])

X0 = np.array([ 0.49019359, -0.11332346,  0.46826879, -0.13540658,  0.37022392, -0.23379722,  0.37143564, -0.2329437 ,  0.37291492, -0.23186138, 0.37469679, -0.23055168,  0.30316716, -0.29125359,  0.30840626, -0.28652415,  0.44230139, -0.16121566,  0.42683712, -0.17683825, 0.32256713, -0.28145402,  0.3280964 , -0.27628293,  0.33245644, -0.27231986,  0.33670266, -0.26854582,  0.2643481 , -0.33007265, 0.27145917, -0.32347124,  0.3864629 , -0.21705415,  0.3808803 , -0.22279507,  0.27458751, -0.32943364,  0.28447461, -0.31990473, 0.2917428 , -0.3130335 ,  0.29848329, -0.30676519,  0.22697144, -0.36744932,  0.2357466 , -0.35918381,  0.32553467, -0.27798238, 0.33200664, -0.27166872,  0.22802673, -0.37599441,  0.24186978, -0.36250956,  0.25182545, -0.35295084,  0.26090483, -0.34434365, 0.19180827, -0.40261249,  0.20193396, -0.39299645,  0.26323078, -0.34028627,  0.28211954, -0.32155583,  0.18444715, -0.419574  , 0.20146085, -0.40291849,  0.21366417, -0.39111212,  0.2247606 , -0.38048788,  0.15946525, -0.43495551,  0.17055441, -0.424376  , 0.20348854, -0.40002851,  0.23321321, -0.37046216,  0.14509726, -0.45892388,  0.16422526, -0.44015407,  0.17807138, -0.42670492, 0.1907319 , -0.41451658,  0.13036714, -0.46405362,  0.14199556, -0.45293485,  0.14977732, -0.45373973,  0.18715638, -0.41651899, 0.11082473, -0.49319641,  0.13088375, -0.47349559,  0.145673  , -0.45910329,  0.15936004, -0.44588844,  0.10475443, -0.48966633, 0.11649699, -0.47843342])
X1 = np.array([ 0.08172583,  0.08172583,  0.12787895,  0.12787895,  0.17680895, 0.17680895,  0.20428698,  0.20428698,  0.22810783,  0.22810783, 0.24952302,  0.24952302,  0.25443032,  0.25443032,  0.27212382, 0.27212382,  0.09939284,  0.09939284,  0.14649492,  0.14649492, 0.18353275,  0.18353275,  0.21186616,  0.21186616,  0.23646753, 0.23646753,  0.25859485,  0.25859485,  0.25241207,  0.25241207, 0.27111512,  0.27111512,  0.11277054,  0.11277054,  0.16042754, 0.16042754,  0.18318121,  0.18318121,  0.21269144,  0.21269144, 0.23825706,  0.23825706,  0.26132525,  0.26132525,  0.24416304, 0.24416304,  0.26402983,  0.26402983,  0.11961642,  0.11961642, 0.16822144,  0.16822144,  0.17599107,  0.17599107,  0.20693342, 0.20693342,  0.23361131,  0.23361131,  0.25782472,  0.25782472, 0.23053159,  0.23053159,  0.2516101 ,  0.2516101 ,  0.11876227, 0.11876227,  0.16908658,  0.16908658,  0.16286772,  0.16286772, 0.19528754,  0.19528754,  0.22310772,  0.22310772,  0.24857796, 0.24857796,  0.21262181,  0.21262181,  0.23482641,  0.23482641, 0.11042389,  0.11042389,  0.16301827,  0.16301827,  0.14522374, 0.14522374,  0.17886349,  0.17886349,  0.20768069,  0.20768069, 0.23437567,  0.23437567,  0.19167763,  0.19167763,  0.21478313, 0.21478313,  0.09612585,  0.09612585,  0.15078275,  0.15078275, 0.1247584 ,  0.1247584 ,  0.15903691,  0.15903691,  0.18850909, 0.18850909,  0.21622738,  0.21622738,  0.16897004,  0.16897004, 0.1926264 ,  0.1926264 ])
X2 = np.array([ 0.0039031 ,  0.0039031 ,  0.00346908,  0.00346908,  0.00450824, 0.00450824,  0.00409751,  0.00409751,  0.0038224 ,  0.0038224 , 0.00358683,  0.00358683,  0.00393648,  0.00393648,  0.00374151, 0.00374151,  0.00488007,  0.00488007,  0.0040774 ,  0.0040774 , 0.00478876,  0.00478876,  0.00434275,  0.00434275,  0.0040458 , 0.0040458 ,  0.00379218,  0.00379218,  0.00397968,  0.00397968, 0.00379608,  0.00379608,  0.00568263,  0.00568263,  0.00457514, 0.00457514,  0.00488406,  0.00488406,  0.00444946,  0.00444946, 0.00415691,  0.00415691,  0.00390482,  0.00390482,  0.00391778, 0.00391778,  0.00375997,  0.00375997,  0.00617576,  0.00617576, 0.00490909,  0.00490909,  0.00478816,  0.00478816,  0.00441244, 0.00441244,  0.00415124,  0.00415124,  0.00392093,  0.00392093, 0.00375961,  0.00375961,  0.00363975,  0.00363975,  0.00627155, 0.00627155,  0.00504258,  0.00504258,  0.00451513,  0.00451513, 0.00423891,  0.00423891,  0.00403303,  0.00403303,  0.00384307, 0.00384307,  0.0035197 ,  0.0035197 ,  0.00344643,  0.00344643, 0.00595365,  0.00595365,  0.00496165,  0.00496165,  0.00409633, 0.00409633,  0.003947  ,  0.003947  ,  0.00381432,  0.00381432, 0.00367948,  0.00367948,  0.00321652,  0.00321652,  0.00319428, 0.00319428,  0.0052817 ,  0.0052817 ,  0.00467728,  0.00467728, 0.00357511,  0.00357511,  0.00356312,  0.00356312,  0.00351338, 0.00351338,  0.0034431 ,  0.0034431 ,  0.00287055,  0.00287055, 0.00289938,  0.00289938])
X = np.array([X0,X1,X2])

The data are such that the solution to the problem Y = X.theta exists, with theta being a vector of d dimension with all 0 and a one at index 54:

>>> Y
array([ 0.24186978,  0.20693342,  0.00441244])
>>> X[0, 54]
0.24186978045754323
>>> X[1, 54]
0.20693341629897405
>>> X[2, 54]
0.0044124449820170455

However when I apply Lasso it is no the expected result ... :

from sklearn.linear_model import Lasso
lasso = Lasso(alpha=0.1)
res = lasso.fit(X,Y)

Giving:

>>> res.coef_.tolist()
[0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, 0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, 0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0]

By changing the penalty coefficient:

lasso = Lasso(alpha=0.01)
res = lasso.fit(X,Y)

The result is still erroneous:

>>> res.coef_.tolist()
  [0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.24488850166974235, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0, 0.0, -0.0] 

How could I retrieve the expected vector of coefficient?

Answer 1

Lasso does not solve the l0-penalized least squares but instead l1-penalized least squares. The solution you get for alpha=0.01 is the Lasso solution (with a single non zero coef of ~0.245 for feature #10).

Even if your solution has a squared reconstruction error of 0.0, it still has a penalty of 1.0 (multiplied by alpha).

The solution for lasso with alpha=1.0 has a small squared reconstruction error of 0.04387 (divided by 2 * n_samples == 6) and a smaller l1 penalty of 0.245 (multiplied by alpha).

The objective function minimized by lasso is given in the docstring:

• http://scikit-learn.org/stable/modules/generated/sklearn.linear_model.Lasso.html

To summarize the different priors (or penalties) commonly used to regularize least squares regression:

• l2 penalty favors any number of non-zero coefficients but with very small absolute values (close to zero)
• l1 penalty favors a small number of non-zero coefficients with small absolute values.
• l0 favors a small number of non zero coefficients of any absolute value.

l0 being non-convex, it is often not as easy to optimize as l1 and l2. This is why people use l1 (lasso) or l1 + l2 (elastic net) in practice to find sparse solutions even if not as clean as l0.