Question

我目前正在高维度的情况下用scikit试验Lasso。标签是Y_i（实数），特征是X_i（X_i是大小为d = 112的向量）。我只有三对（Y_i，X_i）。

d＆gt;＆gt; n = 3所以我们处于高维度。

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])

数据是这样的，即问题Y = X.theta的解决方案存在，其中theta是d维度的向量，全0和索引54处的一个：

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

然而，当我申请Lasso时，这不是预期的结果......：

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

，并提供：

>>> 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]

通过改变惩罚系数：

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

结果仍然是错误的：

>>> 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]

如何检索预期的系数向量？

Answer 1

Lasso没有解决l0 - 惩罚的最小二乘，而是l1 - 惩罚最小二乘。您为alpha=0.01获得的解决方案是Lasso解决方案（对于功能＃10，单个非零系数为~0.245）。

即使您的解决方案的重建误差平方为0.0，它的惩罚仍然为1.0（乘以alpha）。

alpha=1.0套索的解决方案有一个小的平方重建误差0.04387（除以2 * n_samples == 6）和较小的l1惩罚0.245（乘以alpha）

通过套索最小化的目标函数在docstring中给出：

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

总结通常用于规范最小二乘回归的不同先验（或惩罚）：

l2惩罚有利于任意数量的非零系数，但绝对值非常小（接近零）
l1惩罚有利于少量具有小绝对值的非零系数。
l0赞成任意绝对值的少数非零系数。

l0是非凸的，它通常不像l1和l2那样容易优化。这就是人们在实践中使用l1（套索）或l1 + l2（弹性网）来查找稀疏解决方案的原因，即使不像l0那样干净。

用Lasso回归，所有的系数都是0

1 个答案: