Using automatic differentiation (dlgradient) for Euler's Method (gradient descent)

I am trying to use automatic differentiation to approach several optimization problems that I usually solve using gradient descent techniques such as Euler's Method. dlgradient works well as a drop-in solution sometimes, but it sometimes fails even in some simple cases that use only basic arithmetic operations. For instance, I have the following toy problem, which is a much simpler version of a problem I actually want to solve (but still fails):

I have a 2D matrix. I define a binary mask of the same size, and I want to use gradient descent to fill the pixels inside of the mask with 1s (yes, I know I can do this with 1 line of code :) ). I have defined three cost functions (below), and only the first one (cost1) works as I would expect. Is there an issue with cost2 and cost3 that I am missing?

    I = sum((x(:)),1:2);
    cost1 = -sum(x.*msk,1:2);
    cost2 = -sum(abs(x.*msk),1:2);
    cost3 = -sum(x.*msk./I,1:2);

For reference, here is the full code:

    % Use autograd to try to put pixels inside of a mask


% Generate Grids
x = -100:99;
y = x(:);
R = sqrt(x.^2+y.^2);
msk = R<7;
msk = msk-mean(msk(:));
% msk = (msk-.5)*2;

% Random starting guess
randn('seed',0);
zprand = randn(size(msk));

% Starting Guess
zp = dlarray(1*(zprand));

clc
% Loop
mns=[];
mxs=[];
mns = mean(extractdata(zp(:)).*(msk(:)<=0));
mxs = mean(extractdata(zp(:)).*(msk(:)>0));
N = 20;
for i = 1:N
    % Calculate Gradient
    [fval,grad] = dlfeval(@autodiff,zp,msk);
    
    % Follow Gradient
    zp = zp-.25*grad;

    % Store mean within different regions of the mask
    mns(i+1) = mean(extractdata(zp(:)).*(msk(:)<=0));
    mxs(i+1) = mean(extractdata(zp(:)).*(msk(:)>0));

    % Plot
    figure(1);
    imagesc(extractdata(zp));
    colorbar
    title(i)
    drawnow
end

% Plot Progression
figure(2);
plot(1:N+1,mns,1:N+1,mxs)



% Functions
% Gradient
function [y,dydx] = autodiff(zp,msk)
    y = costFunction(zp,msk);
    dydx = dlgradient(y,zp);
end

% Cost Function
function cost1 = costFunction(x,msk)
    I = sum((x(:)),1:2);
    cost1 = -sum(x.*msk,1:2);
    cost2 = -sum(abs(x.*msk),1:2);
    cost3 = -sum(x.*msk./I,1:2);
%     cost4 = sum(abs(zp-msk).^2,1:2);
end

Answer 1

[Altaïr]

Hey @Michael T,

It sounds like the goal is to set the pixels inside the mask to 1 and those outside to 0. Here are a few adjustments that can be made to achieve this:

1. Remove the line msk = msk - mean(msk(:)); as it drives the pixels towards normalized values rather than exactly 1 or 0.

2. Ensure that the means are calculated only for the relevant pixels by changing the following lines:

mns = mean(extractdata(zp(msk <= 0))); % Update mean for pixels outside mask
mxs = mean(extractdata(zp(msk > 0))); % Update mean for pixels inside mask

Update the equations for mxs(i+1) and mns(i+1) accordingly.

3. It's preferable for the cost function to be convex with a unique minimum. The minimum should correspond to the 2D matrix elements matching the desired image msk. An example cost function is:

cost1 = sum((x - msk).^2, 1:2); 

This cost function becomes zero when the current image x matches the desired image msk elementwise.

Also, remember to modify the plotting line for the second figure accordingly. For x defined as an integer array going -10 to 10 (i.e. x=-10:10), the figures are as follows after 20 iterations:

I have attached the modified code for reference. For further verification, the result can be checked using fmincon. Here are some useful links for more information:

• fmincon - https://www.mathworks.com/help/releases/R2021b/optim/ug/fmincon.html
• Related MATLAB Answer: https://www.mathworks.com/matlabcentral/answers/2018856