LASSO近端梯度下降法Proximal Gradient Descent公式推导及代码

LASSO by Proximal Gradient Descent

Prepare:
准备：

from itertools import cycle
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import lasso_path, enet_path
from sklearn import datasets
from copy import deepcopy

X = np.random.randn(100,10)
y = np.dot(X,[1,2,3,4,5,6,7,8,9,10])

Proximal Gradient Descent Framework近端梯度下降算法框架

1. randomly set ${\beta }^{(0)}$ for iteration 0
2. For $k$th iteration:
   ----Compute gradient $\nabla f({\beta }^{(k-1)})$
   ----Set $z={\beta }^{(k-1)}-\frac{1}{L}\nabla f({\beta }^{(k-1)})$
   ----Update ${\beta }^{(k)}=\text{sgn}(z)\cdot \text{max}[|z|-\frac{\lambda }{L},0]$
   ----Check convergence: if yes, end algorithm; else continue update
   Endfor

Here $f(\beta )=\frac{1}{2N}(Y-X\beta {)}^T(Y-X\beta )$, and $\nabla f(\beta )=-\frac{1}{N}{X}^T(Y-X\beta )$,
where the size of $X$, $Y$, $\beta$ is $N\times p$, $N\times 1$, $p\times 1$, which means $N$ samples样本 and $p$ features特征. Parameter $L\ge 1$ can be chosen, and $\frac{1}{L}$ can be considered as step size步长.

Proximal Gradient Descent Details近端梯度下降细节推导

Consider optimization problem:
现考虑优化问题：
$${\text{min}}_{x}f(x)+\lambda \cdot g(x),$$
where $x\in {\mathbb{R}}^{p\times 1}$, $f(x)\in \mathbb{R}$. And $f(x)$ is differentiable convex function, and $g(x)$ is convex but may not differentiable.
$f(x)$是可微凸函数，$g(x)$是凸函数但不一定可微。

For $f(x)$, according to Lipschitz continuity, for $\forall x,y$, there always exists a constant $L$ s.t.
对于$f(x)$，根据利普希茨连续性，对于任意$x,y$，一定存在常数$L$使得满足
$$|f^{\prime }(y)-f^{\prime }(x)|\le L|y-x|.$$
Then this problem can be solved using Proximal Gradient Descent.
可以用近似梯度下降来解决这种问题。

Denote $x^{(k)}$ as the $k$th updated result for $x$, then for $x\to$