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This is my version of explanation. I would suggest you to come up with your own explanation. The best way would be for you to try explain this to somebody else in your own words.

Following is my version of explanation, but this is just an example. You may come up with a better version.

GradientDescent - Two Variables

Convolution would be the one of the most important operations in engineering math. If you don't have any prior study and understanding on how Convolution works, it would not be easy to make sense out of it. For those who is not familiar with Gradient Descent Method, I would suggest you to go through my descriptive note here.

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

< Code 1 >

function main

w1 = -6:.025:6;

w2 = -6:.025:6;

[X,Y] = meshgrid(w1,w2);

Z = -4.0 .* (exp(-(0.5*(X-4).^2 + 0.3*(Y-1/2).^2)) + 2 .* exp(-0.2*((X+2/2).^2 + 0.3*(Y+1/2).^2)))+4;

vx = 3.0;

vy = 4.1;

Lf = 0.2; % Learning Factor

hFig = figure(1,'Position',[300 300 600 500]);

contour(X,Y,Z,25);

hold on;

for i = 1:30

  scale = 1.9;

  [gx1,gx2,gy1,gy2] = GetGradientLineAt_2Var(X,Y,Z,vx,vy,scale)

  plot(vx,vy,'ro','MarkerFaceColor',[1 0 0]);

  if i > 1

    line([gx1 gx2],[gy1 gy2]);

    [gnx,gny] = GetGradientNextAt_2Var(X,Y,Z,vx,vy,Lf)

    plot(gnx,gny,'bo','MarkerFaceColor',[0 0 1]);

    vx = gnx;

    vy = gny;

  end

end;

hold off;

end

function [gnx,gny] = GetGradientNextAt_2Var(x,y,z,vx,vy,Lf)

  [ix,iy] = GetLowerMaxIndex_2Var(x,y,vx,vy);

  [sx,sy] = GetSlopAt_2Var(x,y,z,vx,vy);

  gnx = vx - Lf*sx;

  gny = vy - Lf*sy;

endfunction  

function [gx1,gx2,gy1,gy2] = GetGradientLineAt_2Var(x,y,z,vx,vy,scale)

  [ix,iy] = GetLowerMaxIndex_2Var(x,y,vx,vy);

  [sx,sy] = GetSlopAt_2Var(x,y,z,vx,vy);

  %scale = 0.1;

  gx1 = vx;

  gx2 = vx - scale*sx;

  gy1 = vy;

  gy2 = vy - scale*sy;

endfunction  

function [sx,sy] = GetSlopAt_2Var(x,y,z,vx,vy)

   [i,j] = GetLowerMaxIndex_2Var(x,y,vx,vy);

   x = x(1,:);

   y = y(:,1);

   dx = x(i+1)-x(i);

   dy = y(j+1)-y(j);

   dzx = z(j,i+1)-z(j,i);

   dzy = z(j+1,i)-z(j,i);

   sx = dzx / dx;

   sy = dzy / dy;

endfunction  

function [px,py,pz] = GetPoint3At_2Var(x,y,z,vx,vy)

   [i,j] = GetLowerMaxIndex_2Var(x,y,vx,vy);

   x = x(1,:);

   y = y(:,1);

   px = x(i);

   py = y(i);

   pz = z(j,i);

endfunction  

function [idx,idy] = GetLowerMaxIndex_2Var(x,y,vx,vy)

  idx = 1;

  idy = 1;

  xr = x(1,:);

  yr = y(:,1);

  yr = yr';

  for i = 1:length(xr)

    if xr(i) > vx

       idx = i-1;

       break;

    end;     

  end;

  for j = 1:length(yr)

    if yr(j) > vy

       idy = j-1;

       break;

    end;     

  end;

  %return [idx,idy];

endfunction