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Use Automatic Differentiation In Deep Learning Toolbox

Custom Training and Calculations Using Automatic Differentiation

Automatic differentiation makes it easier to create custom training loops, custom layers, and other deep learning customizations.

Generally, the simplest way to customize deep learning training is to create a dlnetwork. Include the layers you want in the network. Then perform training in a custom loop by using some sort of gradient descent, where the gradient is the gradient of the objective function. The objective function can be classification error, cross-entropy, or any other relevant scalar function of the network weights. See List of Functions with dlarray Support.

This example is a high-level version of a custom training loop. Here, f is the objective function, such as loss, and g is the gradient of the objective function with respect to the weights in the network net. The update function represents some type of gradient descent.

% High-level training loop
n = 1;
while (n < nmax)
    [f,g] = dlfeval(@model,net,dlX,t);
    net = update(net,g);
    n = n + 1;
end

You call dlfeval to compute the numeric value of the objective and gradient. To enable the automatic computation of the gradient, the data dlX must be a dlarray.

dlX = dlarray(X);

The objective function has a dlgradient call to calculate the gradient. The dlgradient call must be inside of the function that dlfeval evaluates.

function [f,g] = model(net,dlX,T)
% Calculate objective using supported functions for dlarray
    y = forward(net,dlX);
    f = fcnvalue(y,T); % crossentropy or similar
    g = dlgradient(f,net.Learnables); % Automatic gradient
end

For an example using a dlnetwork with a simple dlfeval-dlgradient-dlarray syntax, see Grad-CAM Reveals the Why Behind Deep Learning Decisions. For a more complex example using a custom training loop, see Train Generative Adversarial Network (GAN). For further details on custom training using automatic differentiation, see Define Custom Training Loops, Loss Functions, and Networks.

Use dlgradient and dlfeval Together for Automatic Differentiation

To use automatic differentiation, you must call dlgradient inside a function and evaluate the function using dlfeval. Represent the point where you take a derivative as a dlarray object, which manages the data structures and enables tracing of evaluation. For example, the Rosenbrock function is a common test function for optimization.

function [f,grad] = rosenbrock(x)

f = 100*(x(2) - x(1).^2).^2 + (1 - x(1)).^2;
grad = dlgradient(f,x);

end

Calculate the value and gradient of the Rosenbrock function at the point x0 = [–1,2]. To enable automatic differentiation in the Rosenbrock function, pass x0 as a dlarray.

x0 = dlarray([-1,2]);
[fval,gradval] = dlfeval(@rosenbrock,x0)
fval =

  1x1 dlarray

   104


gradval =

  1x2 dlarray

   396   200

For an example using automatic differentiation, see Grad-CAM Reveals the Why Behind Deep Learning Decisions.

Derivative Trace

To evaluate a gradient numerically, a dlarray constructs a data structure for reverse mode differentiation, as described in Automatic Differentiation Background. This data structure is the trace of the derivative computation. Keep in mind these guidelines when using automatic differentiation and the derivative trace:

  • Do not introduce a new dlarray inside of an objective function calculation and attempt to differentiate with respect to that object. For example:

    function [dy,dy1] = fun(x1)
    x2 = dlarray(0);
    y = x1 + x2;
    dy = dlgradient(y,x2); % Error: x2 is untraced
    dy1 = dlgradient(y,x1); % No error even though y has an untraced portion
    end
  • Do not use extractdata with a traced argument. Doing so breaks the tracing. For example:

    fun = @(x)dlgradient(x + atan(extractdata(x)),x);
    % Gradient for any point is 1 due to the leading 'x' term in fun.
    dlfeval(fun,dlarray(2.5))
    ans =
    
      1x1 dlarray
    
         1

    However, you can use extractdata to introduce a new independent variable from a dependent one.

  • Use only supported functions. See List of Functions with dlarray Support. To use an unsupported function f, try to implement f using supported functions.

Characteristics of Automatic Derivatives

  • You can evaluate gradients using automatic differentiation only for scalar-valued functions. Intermediate calculations can have any number of variables, but the final function value must be scalar. If you need to take derivatives of a vector-valued function, take derivatives of one component at a time. In this case, consider setting the dlgradient 'RetainData' name-value pair argument to true.

  • A call to dlgradient evaluates derivatives at a particular point. The software generally makes an arbitrary choice for the value of a derivative when there is no theoretical value. For example, the relu function, relu(x) = max(x,0), is not differentiable at x = 0. However, dlgradient returns a value for the derivative.

    x = dlarray(0);
    y = dlfeval(@(t)dlgradient(relu(t),t),x)
    y =
    
      1x1 dlarray
    
         0

    The value at the nearby point eps is different.

    x = dlarray(eps);
    y = dlfeval(@(t)dlgradient(relu(t),t),x)
    y =
    
      1x1 dlarray
    
         1
  • Currently, dlarray does not allow higher order derivatives. In other words, you cannot calculate a second derivative by calling dlgradient twice.

See Also

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