dlgradient

Compute gradients for custom training loops using automatic differentiation

Syntax

[dydx1,...,dydxk] = dlgradient(y,x1,...,xk)

[dydx1,...,dydxk] = dlgradient(y,x1,...,xk,Name,Value)

Description

The dlgradient function computes derivatives using automatic differentiation.

Tip

For most deep learning tasks, you can use a pretrained neural network and adapt it to your own data. For an example showing how to use transfer learning to retrain a convolutional neural network to classify a new set of images, see Retrain Neural Network to Classify New Images. Alternatively, you can create and train neural networks from scratch using the trainnet and trainingOptions functions.

If the trainingOptions function does not provide the training options that you need for your task, then you can create a custom training loop using automatic differentiation. To learn more, see Train Network Using Custom Training Loop.

If the trainnet function does not provide the loss function that you need for your task, then you can specify a custom loss function to the trainnet as a function handle. For loss functions that require more inputs than the predictions and targets (for example, loss functions that require access to the neural network or additional inputs), train the model using a custom training loop. To learn more, see Train Network Using Custom Training Loop.

If Deep Learning Toolbox™ does not provide the layers you need for your task, then you can create a custom layer. To learn more, see Define Custom Deep Learning Layers. For models that cannot be specified as networks of layers, you can define the model as a function. To learn more, see Train Network Using Model Function.

For more information about which training method to use for which task, see Train Deep Learning Model in MATLAB.

[dydx1,...,dydxk] = dlgradient(y,x1,...,xk) returns the gradients of y with respect to the variables x1 through xk.

Call dlgradient from inside a function passed to dlfeval. See Compute Gradient Using Automatic Differentiation and Use Automatic Differentiation In Deep Learning Toolbox.

[dydx1,...,dydxk] = dlgradient(y,x1,...,xk,Name,Value) returns the gradients and specifies additional options using one or more name-value pairs. For example, dydx = dlgradient(y,x,'RetainData',true) causes the gradient to retain intermediate values for reuse in subsequent dlgradient calls. This syntax can save time, but uses more memory. For more information, see Tips.

Examples

Compute Gradient Using Automatic Differentiation

Rosenbrock's function is a standard test function for optimization. The rosenbrock.m helper function computes the function value and uses automatic differentiation to compute its gradient.

type rosenbrock.m

function [y,dydx] = rosenbrock(x)

y = 100*(x(2) - x(1).^2).^2 + (1 - x(1)).^2;
dydx = dlgradient(y,x);

end

To evaluate Rosenbrock's function and its gradient at the point [–1,2], create a dlarray of the point and then call dlfeval on the function handle @rosenbrock.

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

fval = 
  1×1 dlarray

   104

gradval = 
  1×2 dlarray

   396   200

Alternatively, define Rosenbrock's function as a function of two inputs, x1 and x2.

type rosenbrock2.m

function [y,dydx1,dydx2] = rosenbrock2(x1,x2)

y = 100*(x2 - x1.^2).^2 + (1 - x1).^2;
[dydx1,dydx2] = dlgradient(y,x1,x2);

end

Call dlfeval to evaluate rosenbrock2 on two dlarray arguments representing the inputs –1 and 2.

x1 = dlarray(-1);
x2 = dlarray(2);
[fval,dydx1,dydx2] = dlfeval(@rosenbrock2,x1,x2)

fval = 
  1×1 dlarray

   104

dydx1 = 
  1×1 dlarray

   396

dydx2 = 
  1×1 dlarray

   200

Plot the gradient of Rosenbrock's function for several points in the unit square. First, initialize the arrays representing the evaluation points and the output of the function.

[X1 X2] = meshgrid(linspace(0,1,10));
X1 = dlarray(X1(:));
X2 = dlarray(X2(:));
Y = dlarray(zeros(size(X1)));
DYDX1 = Y;
DYDX2 = Y;

Evaluate the function in a loop. Plot the result using quiver.

for i = 1:length(X1)
    [Y(i),DYDX1(i),DYDX2(i)] = dlfeval(@rosenbrock2,X1(i),X2(i));
end
quiver(extractdata(X1),extractdata(X2),extractdata(DYDX1),extractdata(DYDX2))
xlabel('x1')
ylabel('x2')

Compute Gradients Involving Complex Numbers

Use dlgradient and dlfeval to compute the value and gradient of a function that involves complex numbers. You can compute complex gradients, or restrict the gradients to real numbers only.

Define the function complexFun, listed at the end of this example. This function implements the following complex formula:

$\mathit{f}\left(\mathit{x}\right)=\left(2+3\mathit{i}\right)\mathit{x}$

Define the function gradFun, listed at the end of this example. This function calls complexFun and uses dlgradient to calculate the gradient of the result with respect to the input. For automatic differentiation, the value to differentiate — i.e., the value of the function calculated from the input — must be a real scalar, so the function takes the sum of the real part of the result before calculating the gradient. The function returns the real part of the function value and the gradient, which can be complex.

Define the sample points over the complex plane between -2 and 2 and -2$\mathit{i}$ and 2$\mathit{i}$ and convert to dlarray.

functionRes = linspace(-2,2,100);
x = functionRes + 1i*functionRes.';
x = dlarray(x);

Calculate the function value and gradient at each sample point.

[y, grad] = dlfeval(@gradFun,x);
y = extractdata(y);

Define the sample points at which to display the gradient.

gradientRes = linspace(-2,2,11);
xGrad = gradientRes + 1i*gradientRes.';

Extract the gradient values at these sample points.

[~,gradPlot] = dlfeval(@gradFun,dlarray(xGrad));
gradPlot = extractdata(gradPlot);

Plot the results. Use imagesc to show the value of the function over the complex plane. Use quiver to show the direction and magnitude of the gradient.

imagesc([-2,2],[-2,2],y);
axis xy
colorbar
hold on
quiver(real(xGrad),imag(xGrad),real(gradPlot),imag(gradPlot),"k");
xlabel("Real")
ylabel("Imaginary")
title("Real Value and Gradient","Re$(f(x)) = $ Re$((2+3i)x)$","interpreter","latex")

The gradient of the function is the same across the entire complex plane. Extract the value of the gradient calculated by automatic differentiation.

grad(1,1)

ans = 
  1×1 dlarray

   2.0000 - 3.0000i

By inspection, the complex derivative of the function has the value

$\frac{\mathrm{df}\left(\mathit{x}\right)}{\mathrm{dx}}=2+3\mathit{i}$

However, the function Re($\mathit{f}\left(\mathit{x}\right)$) is not analytic, and therefore no complex derivative is defined. For automatic differentiation in MATLAB, the value to differentiate must always be real, and therefore the function can never be complex analytic. Instead, the derivative is computed such that the returned gradient points in the direction of steepest ascent, as seen in the plot. This is done by interpreting the function Re$\left(\mathit{f}\left(\mathit{x}\right)\right)$: C $\to$ R as a function Re$\left(\mathit{f}\left({\mathit{x}}_{\mathit{R}}+\mathit{i}{\mathit{x}}_{\mathit{I}}\right)\right)$: R $\times$ R $\to$ R.

function y = complexFun(x)
    y = (2+3i)*x;    
end

function [y,grad] = gradFun(x)
    y = complexFun(x);
    y = real(y);

    grad = dlgradient(sum(y,"all"),x);
end

Input Arguments

y — Variable to differentiate

scalar dlarray object

Variable to differentiate, specified as a scalar dlarray object. For differentiation, y must be a traced function of dlarray inputs (see Traced dlarray) and must consist of supported functions for dlarray (see List of Functions with dlarray Support).

Variable to differentiate must be real even when the name-value option 'AllowComplex' is set to true.

Example: 100*(x(2) - x(1).^2).^2 + (1 - x(1)).^2

Example: relu(X)

Data Types: single | double | logical

x1,...,xk — Variable in function

dlarray object | cell array containing dlarray objects | structure containing dlarray objects | table containing dlarray objects

Variable in the function, specified as a dlarray object, a cell array, structure, or table containing dlarray objects, or any combination of such arguments recursively. For example, an argument can be a cell array containing a cell array that contains a structure containing dlarray objects.

If you specify x1,...,xk as a table, the table must contain the following variables:

• Layer — Layer name, specified as a string scalar.

• Parameter — Parameter name, specified as a string scalar.

• Value — Value of parameter, specified as a cell array containing a dlarray.

Example: dlarray([1 2;3 4])

Data Types: single | double | logical | struct | cell
Complex Number Support: Yes

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: dydx = dlgradient(y,x,'RetainData',true) causes the gradient to retain intermediate values for reuse in subsequent dlgradient calls

RetainData — Flag to retain data used to compute gradients

false or 0 (default) | true or 1

Flag to retain data used to compute gradients, specified as one of these values:

• false or 0 — Do not retain data used to compute gradients.

• true or 1 — Retain the data used to compute gradients in dydx1,...,dydxk. Subsequent calls to dlgradient can reuse these values without recomputing them. The software discards these values when the dlfeval function completes the evaluation. This option is useful only when the dlfeval call contains more than one dlgradient function call. It can save time when multiple dlgradient calls use parts of the same trace, at the cost of additional memory usage.

When EnableHigherDerivatives is true, then the software retains the data used to compute gradients and the RetainData argument has no effect.

Example: dydx = dlgradient(y,x,'RetainData',true)

Data Types: logical

EnableHigherDerivatives — Flag to enable higher-order derivatives

true or 1 | false or 0

Flag to enable higher-order derivatives, specified as one of these values:

• Numeric or logical 1 (true) — Enable higher-order derivatives. Trace the backward pass so that the returned values can be used in further computations for subsequent calls to functions that compute derivatives using automatic differentiation (for example, dlgradient, dljacobian, dldivergence, and dllaplacian).

• Numeric or logical 0 (false) — Disable higher-order derivatives. Do not trace the backward pass. When you want to compute only first-order derivatives, this option is usually quicker and requires less memory.

When using the dlgradient function inside an AcceleratedFunction object, the default value is true. Otherwise, the default value is false.

If EnableHigherDerivatives is true, then intermediate values are retained and the RetainData argument has no effect.

For an example that shows how to train a model that require calculating higher-order derivatives, see Train Wasserstein GAN with Gradient Penalty (WGAN-GP).

AllowComplex — Flag to allow complex variables and gradients

true or 1 (default) | false or 0

Flag to allow complex variables in function and complex gradients, specified as one of the following:

• true — Allow complex variables in function and complex gradients. Variables in the function can be specified as complex numbers. Gradients can be complex even if all variables are real. Variable to differentiate must be real.

• false — Do not allow complex variables and gradients. Variable to differentiate and any variables in the function must be real numbers. Gradients are always real. Intermediate values can still be complex.

Variable to differentiate must be real even when the name-value option 'AllowComplex' is set to true.

Data Types: logical

Output Arguments

dydx1,...,dydxk — Gradient

dlarray object | cell array containing dlarray objects | structure containing dlarray objects | table containing dlarray objects

Gradient, returned as a dlarray object, or a cell array, structure, or table containing dlarray objects, or any combination of such arguments recursively. The size and data type of dydx1,...,dydxk are the same as those of the associated input variable x1,…,xk.

Limitations

• The dlgradient function does not support calculating higher-order derivatives when using dlnetwork objects containing custom layers with a custom backward function.

• The dlgradient function does not support calculating higher-order derivatives when using dlnetwork objects containing the following layers:

  • gruLayer

  • lstmLayer

  • bilstmLayer

• The dlgradient function does not support calculating higher-order derivatives that depend on the following functions:

  • gru

  • lstm

  • embed

  • prod

  • interp1

More About

Traced dlarray

During the computation of a function, a dlarray internally records the steps taken in a trace, enabling reverse mode automatic differentiation. The trace occurs within a dlfeval call. See Automatic Differentiation Background.

Tips

• A dlgradient call must be inside a function. To obtain a numeric value of a gradient, you must evaluate the function using dlfeval, and the argument to the function must be a dlarray. See Use Automatic Differentiation In Deep Learning Toolbox.

• To enable the correct evaluation of gradients, the y argument must use only supported functions for dlarray. See List of Functions with dlarray Support.

• If you set the 'RetainData' name-value pair argument to true, the software preserves tracing for the duration of the dlfeval function call instead of erasing the trace immediately after the derivative computation. This preservation can cause a subsequent dlgradient call within the same dlfeval call to be executed faster, but uses more memory. For example, in training an adversarial network, the 'RetainData' setting is useful because the two networks share data and functions during training. See Train Generative Adversarial Network (GAN).

• When you need to calculate first-order derivatives only, ensure that the 'EnableHigherDerivatives' option is false as this is usually quicker and requires less memory.

• Complex gradients are calculated using the Wirtinger derivative. The gradient is defined in the direction of increase of the real part of the function to differentiate. This is because the variable to differentiate — for example, the loss — must be real, even if the function is complex.

• To speed up calls to deep learning functions, such as model functions and model loss functions, you can use the dlaccelerate function. The function returns an AcceleratedFunction object that automatically optimizes, caches, and reuses the traces.

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The dlgradient function supports GPU array input with these usage notes and limitations:

• If the variable to differentiate input argument y is a dlarray object that contains a gpuArray, then this function runs on the GPU.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2019b