关于subGradent descent和Proximal gradient descent的迭代速度

clc;clear;
D=500;N=10000;thre=10e-8;zeroRatio=0.6;
X = randn(N,D);
r=rand(1,D);
r=sign(1-2*r).*(2+2*r);
perm=randperm(D);r(perm(1:floor(D*zeroRatio)))=0;
Y = X*r‘ + randn(N,1)*.1; % small added noise
lamda=1;stepsize=10e-5;
%%% y=x*beta‘
%%% Loss=0.5*(y-x*beta‘)_2++lamda|beta|

%%%% GD
%%% al_y/al_beta=sigma(x_i*(x_i*beta‘-y_i)+fabs(lamda))
beta=zeros(size(r));

pre_error=inf;new_error=0;
while abs(pre_error-new_error)>thre
    pre_error=new_error;
    tmp=0;
    for j=1:length(Y)
        tmp=tmp+X(j,:)*(X(j,:)*beta‘-Y(j,:));
    end
    beta=beta-stepsize*(tmp+lamda);
    new_error=lamda*norm(beta,1);
    for j=1:length(Y)
        new_error=new_error+(Y(j,:)-X(j,:)*beta‘)*(Y(j,:)-X(j,:)*beta‘);
    end
    disp(new_error)
end

% %%%% Proximal GD
% Loss=0.5*(y-x*beta‘)_2++lamda|beta|=g(x)+h(x)
% 左边可导 x_{t+1}=x_{t}-stepsize*sigma(x_i*(x_i*beta‘-y_i)
% X_{t+1}=prox_{l1-norm ball}(x_{t+1})=

disp(‘pgd‘)
beta_pgd=zeros(size(r));
pre_error=inf;new_error=0;
while abs(pre_error-new_error)>thre
    pre_error=new_error;
    tmp=0;
    for j=1:length(Y)
        tmp=tmp+X(j,:)*(X(j,:)*beta_pgd‘-Y(j,:));
    end
    newbeta=beta_pgd-stepsize*(tmp+lamda); add=stepsize*lamda;
    pidx=newbeta>add;beta_pgd(pidx)=newbeta(pidx)-add;
    zeroidx=newbeta<abs(add);beta_pgd(zeroidx)=0;
    nidx=newbeta+add<0;beta_pgd(nidx)=newbeta(nidx)+add;

    new_error=lamda*norm(beta,1);
    for j=1:length(Y)
        new_error=new_error+(Y(j,:)-X(j,:)*beta‘)*(Y(j,:)-X(j,:)*beta‘);
    end
    disp(new_error)
end

PGD的代码说明见下图

PGD主要是project那一步有解析解，速度快

subGradent收敛速度O(1/sqrt(T))