Initialize Learnable Parameters for Model Functions

When training a network using layers, layer graphs, or dlnetwork objects, the software automatically initializes the learnable parameters according to the layer initialization properties. When defining a deep learning model as a function, you must initialize the learnable parameters manually.

The initialization of learnable parameters (for example, weights and biases) can have a big impact on how quickly a deep learning model converges.

Tip

This page explains how to initialize learnable parameters for deep learning models defined as functions in a custom training loop. To learn how to specify the learnable parameter initialization for deep learning layers use the corresponding layer properties. For example, to set the weights initializer of a convolution2dLayer object, use the WeightsInitializer property.

Default Layer Initializations

This table shows the default initializations for the learnable parameters for each layer, and provides links that show how to initialize learnable parameters for model functions using the same initialization.

Layer	Learnable Parameter	Default Initialization
`convolution2dLayer`	Weights	Glorot Initialization
`convolution2dLayer`	Bias	Zeros Initialization
`convolution3dLayer`	Weights	Glorot Initialization
`convolution3dLayer`	Bias	Zeros Initialization
`groupedConvolution2dLayer`	Weights	Glorot Initialization
`groupedConvolution2dLayer`	Bias	Zeros Initialization
`transposedConv2dLayer`	Weights	Glorot Initialization
`transposedConv2dLayer`	Bias	Zeros Initialization
`transposedConv3dLayer`	Weights	Glorot Initialization
`transposedConv3dLayer`	Bias	Zeros Initialization
`fullyConnectedLayer`	Weights	Glorot Initialization
`fullyConnectedLayer`	Bias	Zeros Initialization
`batchNormalizationLayer`	Offset	Zeros Initialization
`batchNormalizationLayer`	Scale	Ones Initialization
`lstmLayer`	Input weights	Glorot Initialization
	Recurrent weights	Orthogonal Initialization
	Bias	Unit Forget Gate Initialization
`gruLayer`	Input weights	Glorot Initialization
	Recurrent weights	Orthogonal Initialization
	Bias	Zeros Initialization
`wordEmbeddingLayer`	Weights	Gaussian Initialization, with mean 0 and standard deviation 0.01.

Learnable Parameter Sizes

When initializing learnable parameters for model functions, you must specify parameters of the correct size. The size of the learnable parameters depends on the type of deep learning operation.

Operation	Learnable Parameter	Size
`batchnorm`	Offset	`[numChannels 1]`, where `numChannels` is the number of input channels.
`batchnorm`	Scale	`[numChannels 1]`, where `numChannels` is the number of input channels.
`dlconv`	Weights	`[filterSize numChannels numFilters]`, where `filterSize` is a 1-by-`K` vector specifying the filter size, `numChannels` is the number of input channels, `numFilters` is the number of filters, and `K` is the number of spatial dimensions.
`dlconv`	Bias	One of the following: `[numFilters 1]`, where `numFilters` is the number of filters `[1 1]`
`dlconv` (grouped)	Weights	`[filterSize numChannelsPerGroup numFiltersPerGroup numGroups]`, where `filterSize` is a 1-by-`K` vector specifying the filter size, `numChannelsPerGroup` is the number of input channels for each group, `numFiltersPerGroup` is the number of filters for each group, `numGroups` is the number of groups, and `K` is the number of spatial dimensions.
`dlconv` (grouped)	Bias	One of the following: `[numFiltersPerGroup 1]`, where `numFiltersPerGroup` is the number of filters for each group. `[1 1]`
`dltranspconv`	Weights	`[filterSize numFilters numChannels]`, where `filterSize` is a 1-by-`K` vector specifying the filter size, `numChannels` is the number of input channels, `numFilters` is the number of filters, and `K` is the number of spatial dimensions.
`dltranspconv`	Bias	One of the following: `[numFilters 1]`, where `numFilters` is the number of filters for each group. `[1 1]`
`dltranspconv` (grouped)	Weights	`[filterSize numFiltersPerGroup numChannelsPerGroup numGroups]`, where `filterSize` is a 1-by-`K` vector specifying the filter size, `numChannelsPerGroup` is the number of input channels for each group, `numFiltersPerGroup` is the number of filters for each group, `numGroups` is the number of groups, and `K` is the number of spatial dimensions.
`dltranspconv` (grouped)	Bias	One of the following: `[numFiltersPerGroup 1]`, where `numFiltersPerGroup` is the number of filters for each group. `[1 1]`
`fullyconnect`	Weights	`[outputSize inputSize]`, where `outputSize` and `inputSize` is the number of output and input channels, respectively.
`fullyconnect`	Bias	`[outputSize 1]`, where `outputSize` is the number of output channels.
`gru`	Input weights	`[3*numHiddenUnits inputSize]`, where `numHiddenUnits` is the number of hidden units of the operation and `inputSize` is the number of input channels.
	Recurrent weights	`[3*numHiddenUnits numHiddenUnits]`, where `numHiddenUnits` is the number of hidden units of the operation.
	Bias	`[3*numHiddenUnits 1]`, where `numHiddenUnits` is the number of hidden units of the operation.
`lstm`	Input weights	`[4*numHiddenUnits inputSize]`, where `numHiddenUnits` is the number of hidden units of the operation and `inputSize` is the number of input channels.
	Recurrent weights	`[4*numHiddenUnits numHiddenUnits]`, where `numHiddenUnits` is the number of hidden units of the operation.
	Bias	`[4*numHiddenUnits 1]`, where `numHiddenUnits` is the number of hidden units of the operation.

Glorot Initialization

The Glorot (also known as Xavier) initializer [1] samples weights from the uniform distribution with bounds $[- \sqrt{\frac{6}{N_{o} + N_{i}}}, \sqrt{\frac{6}{N_{o} + N_{i}}}]$ , where the values of N_o and N_i depend on the type of deep learning operation:

Operation	Learnable Parameter	N_o	N_i
`dlconv`	Weights	`prod(filterSize)*numFilters`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numFilters` is the number of filters, and `K` is the number of spatial dimensions.	`prod(filterSize)*numChannels`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numChannels` is the number of input channels, and `K` is the number of spatial dimensions.
`dlconv` (grouped)	Weights	`prod(filterSize)*numFiltersPerGroup`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numFiltersPerGroup` is the number of filters for each group, and `K` is the number of spatial dimensions.	`prod(filterSize)*numChannelsPerGroup`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numChannelsPerGroup` is the number of input channels for each group, and `K` is the number of spatial dimensions.
`dltranspconv`	Weights	`prod(filterSize)*numFilters`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numFilters` is the number of filters, and `K` is the number of spatial dimensions.	`prod(filterSize)*numChannels`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numChannels` is the number of input channels, and `K` is the number of spatial dimensions.
`dltranspconv` (grouped)	Weights	`prod(filterSize)*numFiltersPerGroup`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numFiltersPerGroup` is the number of filters for each group, and `K` is the number of spatial dimensions.	`prod(filterSize)*numChannelsPerGroup`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numChannelsPerGroup` is the number of input channels for each group, and `K` is the number of spatial dimensions.
`fullyconnect`	Weights	The number of output channels of the operation	The number of input channels of the operation
`gru`	Input weights	`3*numHiddenUnits`, where `numHiddenUnits` is the number of hidden units of the operation.	The number of input channels of the operation
`gru`	Recurrent weights	`3*numHiddenUnits`, where `numHiddenUnits` is the number of hidden units of the operation.	The number of hidden units of the operation.
`lstm`	Input weights	`4*numHiddenUnits`, where `numHiddenUnits` is the number of hidden units of the operation.	The number of input channels of the operation
`lstm`	Recurrent weights	`4*numHiddenUnits`, where `numHiddenUnits` is the number of hidden units of the operation.	The number of hidden units of the operation.

To initialize learnable parameters using the Glorot initializer easily, you can define a custom function. The function initializeGlorot, takes as input the size of the learnable parameters sz and the values N_o and N_i (numOut and numIn, respectively), and returns the sampled weights as a dlarray object with underlying type 'single'.

function weights = initializeGlorot(sz,numOut,numIn)

Z = 2*rand(sz,'single') - 1;
bound = sqrt(6 / (numIn + numOut));

weights = bound * Z;
weights = dlarray(weights);

end

Example

Initialize the weights for a convolutional operation with 128 filters of size 5-by-5 and 3 input channels.

filterSize = [5 5];
numChannels = 3;
numFilters = 128;

sz = [filterSize numChannels numFilters];
numOut = prod(filterSize) * numFilters;
numIn = prod(filterSize) * numFilters;

parameters.conv.Weights = initializeGlorot(sz,numOut,numIn);

He Initialization

The He initializer [44] samples weights from the normal distribution with zero mean and variance $\frac{2}{N_{i}}$ , where the value N_i depends on type of deep learning operation:

Operation	Learnable Parameter	N_i
`dlconv`	Weights	`prod(filterSize)*numChannelsPerGroup`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numChannelsPerGroup` is the number of input channels for each group, and `K` is the number of spatial dimensions.
`dltranspconv`	Weights	`prod(filterSize)*numChannelsPerGroup`, where `filterSize` is a 1-by-`K` vector containing the filter size, `numChannelsPerGroup` is the number of input channels for each group, and `K` is the number of spatial dimensions.
`fullyconnect`	Weights	The number of input channels of the operation
`gru`	Input weights	The number of input channels of the operation
`gru`	Recurrent weights	The number of hidden units of the operation.
`lstm`	Input weights	The number of input channels of the operation
`lstm`	Recurrent weights	The number of hidden units of the operation.

To initialize learnable parameters using the He initializer easily, you can define a custom function. The function initializeHe, takes as input the size of the learnable parameters sz, the value N_i, and returns the sampled weights as a dlarray object with underlying type 'single'.

function weights = initializeHe(sz,numIn)

weights = randn(sz,'single') * sqrt(2/numIn);
weights = dlarray(weights);

end

Example

Initialize the weights for a convolutional operation with 128 filters of size 5-by-5 and 3 input channels.

filterSize = [5 5];
numChannels = 3;
numFilters = 128;

sz = [filterSize numChannels numFilters];
numIn = prod(filterSize) * numFilters;

parameters.conv.Weights = initializeHe(sz,numIn);

Gaussian Initialization

The Gaussian initializer samples weights from a normal distribution.

To initialize learnable parameters using the Gaussian initializer easily, you can define a custom function. The function initializeGaussian, takes as input the size of the learnable parameters sz, the distribution mean mu, the distribution standard deviation sigma, and returns the sampled weights as a dlarray object with underlying type 'single'.

function weights = initializeGaussian(sz,mu,sigma)

weights = randn(sz,'single')*sigma + mu;
weights = dlarray(weights);

end

Example

Initialize the weights for an embedding operation with dimension 300 and vocabulary size 5000 using the Gaussian initializer with mean 0 and standard deviation 0.01.

embeddingDimension = 300;
vocabularySize = 5000;
mu = 0;
sigma = 0.01;

sz = [embeddingDimension vocabularySize];

parameters.emb.Weights = initializeGaussian(sz,mu,sigma);

Uniform Initialization

The uniform initializer samples weights from a uniform distribution.

To initialize learnable parameters using the uniform initializer easily, you can define a custom function. The function initializeUniform, takes as input the size of the learnable parameters sz, the distribution bound bound, and returns the sampled weights as a dlarray object with underlying type 'single'.

function parameter = initializeUniform(sz,bound)

Z = 2*rand(sz,'single') - 1;
parameter = bound * Z;
parameter = dlarray(parameter);

end

Example

Initialize the weights for an attention mechanism with size 100-by-100 and bound 0.1 using the uniform initializer.

sz = [100 100];
bound = 0.1;

parameters.attentionn.Weights = initializeUniform(sz,bound);

Orthogonal Initialization

The orthogonal initializer returns the orthogonal matrix Q given by the QR decomposition of Z = QR, where Z is sampled from a unit normal distribution and the size of Z matches the size of the learnable parameter.

To initialize learnable parameters using the orthogonal initializer easily, you can define a custom function. The function initializeOrthogonal, takes as input the size of the learnable parameters sz and returns the orthogonal matrix as a dlarray object with underlying type 'single'.

function parameter = initializeOrthogonal(sz)

Z = randn(sz,'single');
[Q,R] = qr(Z,0);

D = diag(R);
Q = Q * diag(D ./ abs(D));

parameter = dlarray(Q);

end

Example

Initialize the recurrent weights for an LSTM operation with 100 hidden units using the orthogonal initializer.

numHiddenUnits = 100;

sz = [4*numHiddenUnits numHiddenUnits];

parameters.lstm.RecurrentWeights = initializeOrthogonal(sz);

Unit Forget Gate Initialization

The unit forget gate initializer initializes the bias for an LSTM operation such that the forget gate component of the biases are ones and the remaining entries are zeros.

To initialize learnable parameters using the orthogonal initializer easily, you can define a custom function. The function initializeUnitForgetGate, takes as input the number of hidden units in the LSTM operation and returns the bias as a dlarray object with underlying type 'single'.

function bias = initializeUnitForgetGate(numHiddenUnits)

bias = zeros(4*numHiddenUnits,1,'single');

idx = numHiddenUnits+1:2*numHiddenUnits;
bias(idx) = 1;

bias = dlarray(bias);

end

Example

Initialize the bias of an LSTM operation with 100 hidden units using the unit forget gate initializer.

numHiddenUnits = 100;

parameters.lstm.Bias = initializeUnitForgetGate(numHiddenUnits,'single');

Ones Initialization

To initialize learnable parameters with ones easily, you can define a custom function. The function initializeOnes, takes as input the size of the learnable parameters sz and returns parameters as a dlarray object with underlying type 'single'.

function parameter = initializeOnes(sz)

parameter = ones(sz,'single');
parameter = dlarray(weights);

end

Example

Initialize the scale for a batch normalization operation with 128 input channels with ones.

numChannels = 128;

sz = [numChannels 1];

parameters.bn.Scale = initializeOnes(sz);

Zeros Initialization

To initialize learnable parameters with zeros easily, you can define a custom function. The function initializeZeros, takes as input the size of the learnable parameters sz and returns parameters as a dlarray object with underlying type 'single'.

function parameter = initializeZeros(sz)

parameter = zeros(sz,'single');
parameter = dlarray(weights);

end

Example

Initialize the offset for a batch normalization operation with 128 input channels with zeros.

numChannels = 128;

sz = [numChannels 1];

parameters.bn.Offset = initializeZeros(sz);

References

[1] Glorot, Xavier, and Yoshua Bengio. "Understanding the difficulty of training deep feedforward neural networks." In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pp. 249-256. 2010.

[1] He, Kaiming, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. "Delving deep into rectifiers: Surpassing human-level performance on imagenet classification." In Proceedings of the IEEE international conference on computer vision, pp. 1026-1034. 2015.

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