sgdmupdate

Update parameters using stochastic gradient descent with momentum (SGDM)

Syntax

[netUpdated,vel] = sgdmupdate(net,grad,vel)

[params,vel] = sgdmupdate(params,grad,vel)

[___] = sgdmupdate(___learnRate,momentum)

Description

Update the network learnable parameters in a custom training loop using the stochastic gradient descent with momentum (SGDM) algorithm.

Note

This function applies the SGDM optimization algorithm to update network parameters in custom training loops. To train a neural network using the trainnet function using the SGDM solver, use the trainingOptions function and set the solver to "sgdm".

[netUpdated,vel] = sgdmupdate(net,grad,vel) updates the learnable parameters of the network net using the SGDM algorithm. Use this syntax in a training loop to iteratively update a network defined as a dlnetwork object.

[params,vel] = sgdmupdate(params,grad,vel) updates the learnable parameters in params using the SGDM algorithm. Use this syntax in a training loop to iteratively update the learnable parameters of a network defined using functions.

[___] = sgdmupdate(___learnRate,momentum) also specifies values to use for the global learning rate and momentum, in addition to the input arguments in previous syntaxes.

Examples

Update Learnable Parameters Using sgdmupdate

Perform a single SGDM update step with a global learning rate of 0.05 and momentum of 0.95.

Create the parameters and parameter gradients as numeric arrays.

params = rand(3,3,4);
grad = ones(3,3,4);

Initialize the parameter velocities for the first iteration.

vel = [];

Specify custom values for the global learning rate and momentum.

learnRate = 0.05;
momentum = 0.95;

Update the learnable parameters using sgdmupdate.

[params,vel] = sgdmupdate(params,grad,vel,learnRate,momentum);

Train Network Using sgdmupdate

Use sgdmupdate to train a network using the SGDM algorithm.

Load Training Data

Load the digits training data.

[XTrain,TTrain] = digitTrain4DArrayData;
classes = categories(TTrain);
numClasses = numel(classes);

Define Network

Define the network architecture and specify the average image value using the Mean option in the image input layer.

layers = [
    imageInputLayer([28 28 1],'Mean',mean(XTrain,4))
    convolution2dLayer(5,20)
    reluLayer
    convolution2dLayer(3,20,'Padding',1)
    reluLayer
    convolution2dLayer(3,20,'Padding',1)
    reluLayer
    fullyConnectedLayer(numClasses)
    softmaxLayer];

Create a dlnetwork object from the layer array.

net = dlnetwork(layers);

Define Model Loss Function

Create the helper function modelLoss, listed at the end of the example. The function takes a dlnetwork object and a mini-batch of input data with corresponding labels, and returns the loss and the gradients of the loss with respect to the learnable parameters.

Specify Training Options

Specify the options to use during training.

miniBatchSize = 128;
numEpochs = 20;
numObservations = numel(TTrain);
numIterationsPerEpoch = floor(numObservations./miniBatchSize);

Train Network

Initialize the velocity parameter.

vel = [];

Calculate the total number of iterations for the training progress monitor.

numIterations = numEpochs * numIterationsPerEpoch;

Initialize the TrainingProgressMonitor object. Because the timer starts when you create the monitor object, make sure that you create the object close to the training loop.

monitor = trainingProgressMonitor(Metrics="Loss",Info="Epoch",XLabel="Iteration");

Train the model using a custom training loop. For each epoch, shuffle the data and loop over mini-batches of data. Update the network parameters using the sgdmupdate function. At the end of each iteration, display the training progress.

Train on a GPU, if one is available. Using a GPU requires Parallel Computing Toolbox™ and a supported GPU device. For information on supported devices, see GPU Computing Requirements (Parallel Computing Toolbox).

iteration = 0;
epoch = 0;

while epoch < numEpochs && ~monitor.Stop
    epoch = epoch + 1;

    % Shuffle data.
    idx = randperm(numel(TTrain));
    XTrain = XTrain(:,:,:,idx);
    TTrain = TTrain(idx);

    i = 0;
    while i < numIterationsPerEpoch && ~monitor.Stop
        i = i + 1;
        iteration = iteration + 1;

        % Read mini-batch of data and convert the labels to dummy
        % variables.
        idx = (i-1)*miniBatchSize+1:i*miniBatchSize;
        X = XTrain(:,:,:,idx);

        T = zeros(numClasses, miniBatchSize,"single");
        for c = 1:numClasses
            T(c,TTrain(idx)==classes(c)) = 1;
        end

        % Convert mini-batch of data to a dlarray.
        X = dlarray(single(X),"SSCB");

        % If training on a GPU, then convert data to a gpuArray.
        if canUseGPU
            X = gpuArray(X);
        end

        % Evaluate the model loss and gradients using dlfeval and the
        % modelLoss function.
        [loss,gradients] = dlfeval(@modelLoss,net,X,T);

        % Update the network parameters using the SGDM optimizer.
        [net,vel] = sgdmupdate(net,gradients,vel);

        % Update the training progress monitor.
        recordMetrics(monitor,iteration,Loss=loss);
        updateInfo(monitor,Epoch=epoch + " of " + numEpochs);
        monitor.Progress = 100 * iteration/numIterations;
    end
end

Test the Network

Test the classification accuracy of the model by comparing the predictions on a test set with the true labels.

[XTest,TTest] = digitTest4DArrayData;

Convert the data to a dlarray with the dimension format "SSCB" (spatial, spatial, channel, batch). For GPU prediction, also convert the data to a gpuArray.

XTest = dlarray(XTest,"SSCB");
if canUseGPU
    XTest = gpuArray(XTest);
end

To classify images using a dlnetwork object, use the predict function and find the classes with the highest scores.

YTest = predict(net,XTest);
[~,idx] = max(extractdata(YTest),[],1);
YTest = classes(idx);

Evaluate the classification accuracy.

accuracy = mean(YTest==TTest)

accuracy = 0.9910

Model Loss Function

The modelLoss function takes a dlnetwork object net and a mini-batch of input data X with corresponding labels T, and returns the loss and the gradients of the loss with respect to the learnable parameters in net. To compute the gradients automatically, use the dlgradient function.

function [loss,gradients] = modelLoss(net,X,T)

Y = forward(net,X);

loss = crossentropy(Y,T);

gradients = dlgradient(loss,net.Learnables);

end

Input Arguments

net — Network

dlnetwork object

Network, specified as a dlnetwork object.

The function updates the Learnables property of the dlnetwork object. net.Learnables is a table with three 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.

The input argument grad must be a table of the same form as net.Learnables.

params — Network learnable parameters

dlarray | numeric array | cell array | structure | table

Network learnable parameters, specified as a dlarray, a numeric array, a cell array, a structure, or a table.

If you specify params as a table, it must contain the following three 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.

You can specify params as a container of learnable parameters for your network using a cell array, structure, or table, or nested cell arrays or structures. The learnable parameters inside the cell array, structure, or table must be dlarray or numeric values of data type double or single.

The input argument grad must be provided with exactly the same data type, ordering, and fields (for structures) or variables (for tables) as params.

The learnables can be complex-valued. (since R2024a) Ensure that the corresponding operations support complex-valued learnables.

Before R2024a: The learnables must not be complex-valued. If your model involves complex learnables, then convert the learnables to real values before calculating the gradients.

Data Types: single | double | struct | table | cell

grad — Gradients of the loss

dlarray | numeric array | cell array | structure | table

Gradients of the loss, specified as a dlarray, a numeric array, a cell array, a structure, or a table.

The exact form of grad depends on the input network or learnable parameters. The following table shows the required format for grad for possible inputs to sgdmupdate.

Input	Learnable Parameters	Gradients
net	Table net.Learnables containing Layer, Parameter, and Value variables. The Value variable consists of cell arrays that contain each learnable parameter as a dlarray.	Table with the same data type, variables, and ordering as net.Learnables. grad must have a Value variable consisting of cell arrays that contain the gradient of each learnable parameter.
params	dlarray	dlarray with the same data type and ordering as params
	Numeric array	Numeric array with the same data type and ordering as params
	Cell array	Cell array with the same data types, structure, and ordering as params
	Structure	Structure with the same data types, fields, and ordering as params
	Table with Layer, Parameter, and Value variables. The Value variable must consist of cell arrays that contain each learnable parameter as a dlarray.	Table with the same data types, variables, and ordering as params. grad must have a Value variable consisting of cell arrays that contain the gradient of each learnable parameter.

You can obtain grad from a call to dlfeval that evaluates a function that contains a call to dlgradient. For more information, see Use Automatic Differentiation In Deep Learning Toolbox.

The gradients can be complex-valued. (since R2024a) Using complex valued gradients can lead to complex-valued learnable parameters. Ensure that the corresponding operations support complex-valued learnables.

Before R2024a: The gradients must not be complex-valued. If your model involves complex numbers, then convert all outputs to real values before calculating the gradients.

vel — Parameter velocities

[] | dlarray | numeric array | cell array | structure | table

Parameter velocities, specified as an empty array, a dlarray, a numeric array, a cell array, a structure, or a table.

The exact form of vel depends on the input network or learnable parameters. The following table shows the required format for vel for possible inputs to sgdmpdate.

Input	Learnable Parameters	Velocities
net	Table net.Learnables containing Layer, Parameter, and Value variables. The Value variable consists of cell arrays that contain each learnable parameter as a dlarray.	Table with the same data type, variables, and ordering as net.Learnables. vel must have a Value variable consisting of cell arrays that contain the velocity of each learnable parameter.
params	dlarray	dlarray with the same data type and ordering as params
	Numeric array	Numeric array with the same data type and ordering as params
	Cell array	Cell array with the same data types, structure, and ordering as params
	Structure	Structure with the same data types, fields, and ordering as params
	Table with Layer, Parameter, and Value variables. The Value variable must consist of cell arrays that contain each learnable parameter as a dlarray.	Table with the same data types, variables, and ordering as params. vel must have a Value variable consisting of cell arrays that contain the velocity of each learnable parameter.

If you specify vel as an empty array, the function assumes no previous velocities and runs in the same way as for the first update in a series of iterations. To update the learnable parameters iteratively, use the vel output of a previous call to sgdmupdate as the vel input.

The velocity can be complex-valued. (since R2024a) Using complex valued gradients and velocities can lead to complex-valued learnable parameters. Ensure that the corresponding operations support complex-valued learnables.

Before R2024a: The gradients must not be complex-valued. If your model involves complex numbers, then convert all outputs to real values before calculating the gradients.

learnRate — Global learning rate

0.01 (default) | positive scalar

Learning rate, specified as a positive scalar. The default value of learnRate is 0.01.

If you specify the network parameters as a dlnetwork object, the learning rate for each parameter is the global learning rate multiplied by the corresponding learning rate factor property defined in the network layers.

momentum — Momentum

0.9 (default) | positive scalar between 0 and 1

Momentum, specified as a positive scalar between 0 and 1. The default value of momentum is 0.9.

Output Arguments

netUpdated — Updated network

dlnetwork object

Updated network, returned as a dlnetwork object.

The function updates the Learnables property of the dlnetwork object.

params — Updated network learnable parameters

dlarray | numeric array | cell array | structure | table

Updated network learnable parameters, returned as a dlarray, a numeric array, a cell array, a structure, or a table with a Value variable containing the updated learnable parameters of the network.

The learnables can be complex-valued. (since R2024a) Ensure that the corresponding operations support complex-valued learnables.

Before R2024a: The learnables must not be complex-valued. If your model involves complex learnables, then convert the learnables to real values before calculating the gradients.

vel — Updated parameter velocities

dlarray | numeric array | cell array | structure | table

Updated parameter velocities, returned as a dlarray, a numeric array, a cell array, a structure, or a table.

Algorithms

Stochastic Gradient Descent

The standard gradient descent algorithm updates the network parameters (weights and biases) to minimize the loss function by taking small steps at each iteration in the direction of the negative gradient of the loss,

$${\theta }_{\ell +1}={\theta }_{\ell }-\alpha \nabla E\left({\theta }_{\ell }\right),$$

where $$\ell$$is the iteration number, $$\alpha >0$$ is the learning rate, $$\theta$$ is the parameter vector, and $$E\left(\theta \right)$$ is the loss function. In the standard gradient descent algorithm, the gradient of the loss function, $$\nabla E\left(\theta \right)$$, is evaluated using the entire training set, and the standard gradient descent algorithm uses the entire data set at once.

By contrast, at each iteration the stochastic gradient descent algorithm evaluates the gradient and updates the parameters using a subset of the training data. A different subset, called a mini-batch, is used at each iteration. The full pass of the training algorithm over the entire training set using mini-batches is one epoch. Stochastic gradient descent is stochastic because the parameter updates computed using a mini-batch is a noisy estimate of the parameter update that would result from using the full data set.

Stochastic Gradient Descent with Momentum

The stochastic gradient descent algorithm can oscillate along the path of steepest descent towards the optimum. Adding a momentum term to the parameter update is one way to reduce this oscillation [1]. The stochastic gradient descent with momentum (SGDM) update is

$${\theta }_{\ell +1}={\theta }_{\ell }-\alpha \nabla E\left({\theta }_{\ell }\right)+\gamma \left({\theta }_{\ell }-{\theta }_{\ell -1}\right),$$

where the learning rate α and the momentum value $$\gamma$$ determine the contribution of the previous gradient step to the current iteration.

References

[1] Murphy, K. P. Machine Learning: A Probabilistic Perspective. The MIT Press, Cambridge, Massachusetts, 2012.

Extended Capabilities

GPU Arrays

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

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

• When at least one of the following input arguments is a gpuArray or a dlarray with underlying data of type gpuArray, this function runs on the GPU.

  • grad

  • params

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

Version History

Introduced in R2019b

R2024a: Complex-valued learnable parameters and gradients

The learnable parameters, gradients, and velocity can be complex-valued. When the updated learnable parameters are complex-valued, ensure that the corresponding operations support complex-valued parameters.