batchNormalizationLayer

Batch normalization layer

Description

A batch normalization layer normalizes a mini-batch of data across all observations for each channel independently. To speed up training of the convolutional neural network and reduce the sensitivity to network initialization, use batch normalization layers between convolutional layers and nonlinearities, such as ReLU layers.

After normalization, the layer scales the input with a learnable scale factor γ and shifts by a learnable offset β.

Creation

Syntax

layer = batchNormalizationLayer

layer = batchNormalizationLayer(Name,Value)

Description

layer = batchNormalizationLayer creates a batch normalization layer.

layer = batchNormalizationLayer(Name,Value) creates a batch normalization layer and sets the optional TrainedMean, TrainedVariance, Epsilon, Parameters and Initialization, Learn Rate and Regularization, and Name properties using one or more name-value pairs. For example, batchNormalizationLayer('Name','batchnorm') creates a batch normalization layer with the name 'batchnorm'.

Properties

Batch Normalization

TrainedMean — Mean statistic used for prediction

numeric array

Mean statistic used for prediction, specified as one of the following:

• For 2-D image input, a numeric array of size 1-by-1-by-NumChannels

• For 3-D image input, a numeric array of size 1-by-1-by-1-by-NumChannels

• For feature or sequence input, a numeric array of size NumChannels-by-1

If the 'BatchNormalizationStatistics' training option is 'moving', then the software approximates the batch normalization statistics during training using a running estimate and, after training, sets the TrainedMean and TrainedVariance properties to the latest values of the moving estimates of the mean and variance, respectively.

If the 'BatchNormalizationStatistics' training option is 'population', then after network training finishes, the software passes through the data once more and sets the TrainedMean and TrainedVariance properties to mean and variance computed from the entire training data set, respectively.

The layer uses the TrainedMean and TrainedVariance to normalize the input during prediction.

TrainedVariance — Variance statistic used for prediction

numeric array

Variance statistic used for prediction, specified as one of the following:

• For 2-D image input, a numeric array of size 1-by-1-by-NumChannels

• For 3-D image input, a numeric array of size 1-by-1-by-1-by-NumChannels

• For feature or sequence input, a numeric array of size NumChannels-by-1

If the 'BatchNormalizationStatistics' training option is 'moving', then the software approximates the batch normalization statistics during training using a running estimate and, after training, sets the TrainedMean and TrainedVariance properties to the latest values of the moving estimates of the mean and variance, respectively.

If the 'BatchNormalizationStatistics' training option is 'population', then after network training finishes, the software passes through the data once more and sets the TrainedMean and TrainedVariance properties to mean and variance computed from the entire training data set, respectively.

The layer uses the TrainedMean and TrainedVariance to normalize the input during prediction.

Epsilon — Constant to add to mini-batch variances

1e-5 (default) | numeric scalar

Constant to add to the mini-batch variances, specified as a numeric scalar equal to or larger than 1e-5.

The layer adds this constant to the mini-batch variances before normalization to ensure numerical stability and avoid division by zero.

NumChannels — Number of input channels

'auto' (default) | positive integer

Number of input channels, specified as 'auto' or a positive integer.

This property is always equal to the number of channels of the input to the layer. If NumChannels equals 'auto', then the software automatically determines the correct value for the number of channels at training time.

Parameters and Initialization

ScaleInitializer — Function to initialize channel scale factors

'ones' (default) | 'narrow-normal' | function handle

Function to initialize the channel scale factors, specified as one of the following:

• 'ones' – Initialize the channel scale factors with ones.

• 'zeros' – Initialize the channel scale factors with zeros.

• 'narrow-normal' – Initialize the channel scale factors by independently sampling from a normal distribution with zero mean and standard deviation 0.01.

• Function handle – Initialize the channel scale factors with a custom function. If you specify a function handle, then the function must be of the form scale = func(sz), where sz is the size of the scale. For an example, see Specify Custom Weight Initialization Function.

The layer only initializes the channel scale factors when the Scale property is empty.

Data Types: char | string | function_handle

OffsetInitializer — Function to initialize channel offsets

'zeros' (default) | 'ones' | 'narrow-normal' | function handle

Function to initialize the channel offsets, specified as one of the following:

• 'zeros' – Initialize the channel offsets with zeros.

• 'ones' – Initialize the channel offsets with ones.

• 'narrow-normal' – Initialize the channel offsets by independently sampling from a normal distribution with zero mean and standard deviation 0.01.

• Function handle – Initialize the channel offsets with a custom function. If you specify a function handle, then the function must be of the form offset = func(sz), where sz is the size of the scale. For an example, see Specify Custom Weight Initialization Function.

The layer only initializes the channel offsets when the Offset property is empty.

Data Types: char | string | function_handle

Scale — Channel scale factors

[] (default) | numeric array

Channel scale factors γ, specified as a numeric array.

The channel scale factors are learnable parameters. When training a network, if Scale is nonempty, then trainNetwork uses the Scale property as the initial value. If Scale is empty, then trainNetwork uses the initializer specified by ScaleInitializer.

At training time, Scale is one of the following:

• For 2-D image input, a numeric array of size 1-by-1-by-NumChannels

• For 3-D image input, a numeric array of size 1-by-1-by-1-by-NumChannels

• For feature or sequence input, a numeric array of size NumChannels-by-1

Offset — Channel offsets

[] (default) | numeric array

Channel offsets β, specified as a numeric array.

The channel offsets are learnable parameters. When training a network, if Offset is nonempty, then trainNetwork uses the Offset property as the initial value. If Offset is empty, then trainNetwork uses the initializer specified by OffsetInitializer.

At training time, Offset is one of the following:

• For 2-D image input, a numeric array of size 1-by-1-by-NumChannels

• For 3-D image input, a numeric array of size 1-by-1-by-1-by-NumChannels

• For feature or sequence input, a numeric array of size NumChannels-by-1

MeanDecay — Decay value for the moving mean computation

0.1 (default) | numeric scalar between 0 and 1

Decay value for the moving mean computation, specified as a numeric scalar between 0 and 1.

When the 'BatchNormalizationStatistics' training option is 'moving', at each iteration, the layer updates the moving mean value using

$${\mu }^{*}={\lambda }_{\mu }\widehat{\mu }+(1-{\lambda }_{\mu })\mu ,$$

where $${\mu }^{*}$$ denotes the updated mean, $${\lambda }_{\mu }$$ denotes the mean decay value, $$\widehat{\mu }$$ denotes the mean of the layer input, and $$\mu$$ denotes the latest value of the moving mean value.

If the 'BatchNormalizationStatistics' training option is 'population', then this option has no effect.

Data Types: single | double

VarianceDecay — Decay value for the moving variance computation

0.1 (default) | numeric scalar between 0 and 1

Decay value for the moving variance computation, specified as a numeric scalar between 0 and 1.

When the 'BatchNormalizationStatistics' training option is 'moving', at each iteration, the layer updates the moving variance value using

$${\sigma }^2{}^{*}={\lambda }_{{\sigma }^2}\widehat{{\sigma }^2}+(1-{\lambda }_{{\sigma }^2}){\sigma }^2,$$

where $${\sigma }^2{}^{*}$$ denotes the updated variance, $${\lambda }_{{\sigma }^2}$$ denotes the variance decay value, $$\widehat{{\sigma }^2}$$ denotes the variance of the layer input, and $${\sigma }^2$$ denotes the latest value of the moving variance value.

If the 'BatchNormalizationStatistics' training option is 'population', then this option has no effect.

Data Types: single | double

Learn Rate and Regularization

ScaleLearnRateFactor — Learning rate factor for scale factors

1 (default) | nonnegative scalar

Learning rate factor for the scale factors, specified as a nonnegative scalar.

The software multiplies this factor by the global learning rate to determine the learning rate for the scale factors in a layer. For example, if ScaleLearnRateFactor is 2, then the learning rate for the scale factors in the layer is twice the current global learning rate. The software determines the global learning rate based on the settings specified with the trainingOptions function.

OffsetLearnRateFactor — Learning rate factor for offsets

1 (default) | nonnegative scalar

Learning rate factor for the offsets, specified as a nonnegative scalar.

The software multiplies this factor by the global learning rate to determine the learning rate for the offsets in a layer. For example, if OffsetLearnRateFactor equals 2, then the learning rate for the offsets in the layer is twice the current global learning rate. The software determines the global learning rate based on the settings specified with the trainingOptions function.

ScaleL2Factor — L₂ regularization factor for scale factors

1 (default) | nonnegative scalar

L₂ regularization factor for the scale factors, specified as a nonnegative scalar.

The software multiplies this factor by the global L₂ regularization factor to determine the learning rate for the scale factors in a layer. For example, if ScaleL2Factor is 2, then the L₂ regularization for the offsets in the layer is twice the global L₂ regularization factor. You can specify the global L₂ regularization factor using the trainingOptions function.

OffsetL2Factor — L₂ regularization factor for offsets

1 (default) | nonnegative scalar

L₂ regularization factor for the offsets, specified as a nonnegative scalar.

The software multiplies this factor by the global L₂ regularization factor to determine the learning rate for the offsets in a layer. For example, if OffsetL2Factor is 2, then the L₂ regularization for the offsets in the layer is twice the global L₂ regularization factor. You can specify the global L₂ regularization factor using the trainingOptions function.

Layer

Name — Layer name

'' (default) | character vector | string scalar

Layer name, specified as a character vector or a string scalar. To include a layer in a layer graph, you must specify a nonempty, unique layer name. If you train a series network with the layer and Name is set to '', then the software automatically assigns a name to the layer at training time.

Data Types: char | string

NumInputs — Number of inputs

1 (default)

Number of inputs of the layer. This layer accepts a single input only.

Data Types: double

InputNames — Input names

{'in'} (default)

Input names of the layer. This layer accepts a single input only.

Data Types: cell

NumOutputs — Number of outputs

1 (default)

Number of outputs of the layer. This layer has a single output only.

Data Types: double

OutputNames — Output names

{'out'} (default)

Output names of the layer. This layer has a single output only.

Data Types: cell

Examples

Create Batch Normalization Layer

Create a batch normalization layer with the name 'BN1'.

layer = batchNormalizationLayer('Name','BN1')

layer = 
  BatchNormalizationLayer with properties:

               Name: 'BN1'
        NumChannels: 'auto'
        TrainedMean: []
    TrainedVariance: []

   Hyperparameters
          MeanDecay: 0.1000
      VarianceDecay: 0.1000
            Epsilon: 1.0000e-05

   Learnable Parameters
             Offset: []
              Scale: []

  Show all properties

Include batch normalization layers in a Layer array.

layers = [
    imageInputLayer([32 32 3]) 
  
    convolution2dLayer(3,16,'Padding',1)
    batchNormalizationLayer
    reluLayer   
    
    maxPooling2dLayer(2,'Stride',2)
    
    convolution2dLayer(3,32,'Padding',1)
    batchNormalizationLayer
    reluLayer
          
    fullyConnectedLayer(10)
    softmaxLayer
    classificationLayer
    ]

layers = 
  11x1 Layer array with layers:

     1   ''   Image Input             32x32x3 images with 'zerocenter' normalization
     2   ''   Convolution             16 3x3 convolutions with stride [1  1] and padding [1  1  1  1]
     3   ''   Batch Normalization     Batch normalization
     4   ''   ReLU                    ReLU
     5   ''   Max Pooling             2x2 max pooling with stride [2  2] and padding [0  0  0  0]
     6   ''   Convolution             32 3x3 convolutions with stride [1  1] and padding [1  1  1  1]
     7   ''   Batch Normalization     Batch normalization
     8   ''   ReLU                    ReLU
     9   ''   Fully Connected         10 fully connected layer
    10   ''   Softmax                 softmax
    11   ''   Classification Output   crossentropyex

More About

Batch Normalization Layer

A batch normalization layer normalizes a mini-batch of data across all observations for each channel independently. To speed up training of the convolutional neural network and reduce the sensitivity to network initialization, use batch normalization layers between convolutional layers and nonlinearities, such as ReLU layers.

The layer first normalizes the activations of each channel by subtracting the mini-batch mean and dividing by the mini-batch standard deviation. Then, the layer shifts the input by a learnable offset β and scales it by a learnable scale factor γ. β and γ are themselves learnable parameters that are updated during network training.

Batch normalization layers normalize the activations and gradients propagating through a neural network, making network training an easier optimization problem. To take full advantage of this fact, you can try increasing the learning rate. Since the optimization problem is easier, the parameter updates can be larger and the network can learn faster. You can also try reducing the L₂ and dropout regularization. With batch normalization layers, the activations of a specific image during training depend on which images happen to appear in the same mini-batch. To take full advantage of this regularizing effect, try shuffling the training data before every training epoch. To specify how often to shuffle the data during training, use the 'Shuffle' name-value pair argument of trainingOptions.

Algorithms

The batch normalization operation normalizes the elements x_i of the input by first calculating the mean μ_B and variance σ_B² over the spatial, time, and observation dimensions for each channel independently. Then, it calculates the normalized activations as

$$\widehat{x_i}=\frac{x_i-{\mu }_B}{\sqrt{{\sigma }_B^2+ϵ}},$$

where ϵ is a constant that improves numerical stability when the variance is very small.

To allow for the possibility that inputs with zero mean and unit variance are not optimal for the operations that follow batch normalization, the batch normalization operation further shifts and scales the activations using the transformation

$$y_i=\gamma {\widehat{x}}_i+\beta ,$$

where the offset β and scale factor γ are learnable parameters that are updated during network training.

To make predictions with the network after training, batch normalization requires a fixed mean and variance to normalize the data. This fixed mean and variance can be calculated from the training data after training, or approximated during training using running statistic computations.

If the 'BatchNormalizationStatistics' training option is 'moving', then the software approximates the batch normalization statistics during training using a running estimate and, after training, sets the TrainedMean and TrainedVariance properties to the latest values of the moving estimates of the mean and variance, respectively.

If the 'BatchNormalizationStatistics' training option is 'population', then after network training finishes, the software passes through the data once more and sets the TrainedMean and TrainedVariance properties to mean and variance computed from the entire training data set, respectively.

The layer uses the TrainedMean and TrainedVariance to normalize the input during prediction.

References

[1] Ioffe, Sergey, and Christian Szegedy. "Batch normalization: Accelerating deep network training by reducing internal covariate shift." preprint, arXiv:1502.03167 (2015).

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