reddiexample.m

function toyproblem()
%maxstep=1000000;
%ave=1;
%xi=zeros(ave,maxstep);

%xi=mean(xi,1);
%bg=0;

%plot(1+bg*maxstep:maxstep,log10(abs(xi(1,1+bg*maxstep:maxstep)+1+10^(-20))),'Color',[0.1 1 0],'LineWidth',1);
%hold on;
%xlabel('number of iteration','Fontsize',12);
%ylabel('lg(x-x^*)','Fontsize',12);
%hold off;
%slope=(log10(abs(xi(maxstep)))-log10(abs(xi(maxstep/10))))/maxstep*10/9;

plot_map()
end

function plot_map()
maxstep=1000000;
a=41;
b=30;
beta2_conv=zeros(a*b,1);
beta2_div=zeros(a*b,1);
C_conv=zeros(a*b,1);
C_div=zeros(a*b,1);
temp=10;
st_c=1;
st_d=1;
    for i=1:40
    for C=2:(b+2)
            beta2=1-10^(-i/temp);
            xi=zeros(maxstep,1);
            [~,~,xi]=reddiexample(maxstep,C,beta2);
            terminal=sum(log10(abs(xi(maxstep*0.75:maxstep)+1)))/maxstep*4;
            if terminal<-2
                beta2_conv(st_c)= i/temp;
                C_conv(st_c)=C;
                st_c=st_c+1;
            else
                beta2_div(st_d)=i/temp;
                C_div(st_d)=C;
                st_d=st_d+1;
            end                
     end
     i
    end
    beta2_conv=beta2_conv(1:st_c);
    C_conv=C_conv(1:st_c);
    beta2_div=beta2_div(1:st_d);
    C_div=C_div(1:st_d);
    plot(beta2_conv,C_conv,'o');
    hold on;
    plot(beta2_div,C_div,'+');
    xlabel('$-\log_{10}(1-\beta_2)$','interpreter','latex','Fontsize',20);
    ylabel('C','Fontsize',20);
    legend('convergent', 'divergent');
    hold off;
saveas(gcf, 'result', 'png')
end

function [vk,gk,xk]=reddiexample(maxstep,C,beta2)
%C=10;
eta=1;
beta1=0;
%beta2=1-1/C^2;
1-1/C^2;

%beta2=0.95;
xk=zeros(maxstep,1);
vk=zeros(maxstep,1);
gk=zeros(maxstep,1);
%grad=0;
momentum=0;
j=0;
xk(1)=100;
grad=xk(1)*C;
(1/C^2)^(2/(C-2));
    for k=1:maxstep
        %i=ceil(3*rand());
        j=j+1;
        i=mod(j,C);
        %i=ceil(C*rand());
        if i==1
            gr=C;
        else
            gr=-1;
        end
        gk(k)=gr^2;
        grad=grad*beta2+(1-beta2)*gr^2;
        vk(k)=grad;
        momentum=momentum*beta1+(1-beta1)*gr;
        xk(k+1)=xk(k)-eta*momentum/sqrt(grad)*(k)^(-0.5);
        if xk(k+1)>1
            xk(k+1)=1;
        end
        if xk(k+1)<-1
            xk(k+1)=-1;
        end            
    end
    xk=xk(1:maxstep);
end