BatchNormForwardTraining

Versioned name: BatchNormForwardTraining-1

Category: Normalization

Short description: BatchNormForwardTraining works on forward pass at training mode.

Attributes:

  • epsilon

    • Description: epsilon is the number to be added to the variance to avoid division by zero when normalizing a value. For example, epsilon equal to 0.001 means that 0.001 is added to the variance.

    • Range of values: a positive floating-point number

    • Type: float

    • Required: yes

  • momentum

    • Description: momentum is used for the computation of running_mean and running_var. If it’s not available, a cumulative moving average (i.e. simple average) will be computed.

    • Range of values: a positive floating-point number

    • Type: float

    • Required: no

  • data_format

    • Description: data_format denotes the data format of the input and output data.

    • Range of values: NXC or NCX (X means HW for 2D, DHW for 3D)

    • Type: string

    • Default value: NXC

    • Required: no

Inputs

  • 1: input - input tensor with data for normalization. The format is specified by data_format. Required.

  • 2: gamma - gamma scaling for normalized value. A 1D tensor of type T with the same span as input’s channel axis. Optional.

  • 3: beta - beta added to the scaled normalized value. A 1D tensor of type T with the same span as input’s channel axis. Optional.

  • 4: mean - value for mean normalization. A 1D tensor of type T with the same span as input’s channel axis. Required.

  • 5: variance - value for variance normalization. A 1D tensor of type T with the same span as input’s channel axis. Required.

Outputs

  • 1: output - the result of normalization. A tensor of the same type, shape and format with 1st input tensor.

  • 2: running mean - the computed running mean.

  • 3: running variance - the computed running variance.

  • 4: batch mean - the computed batch mean.

  • 5: batch variance - the computed batch variance.

Mathematical Formulation

BatchNormForwardTraining normalizes the output in each hidden layer.

  • Input: Values of \(x\) over a mini-batch:

    \[\beta = \{ x_{1...m} \}\]
  • Parameters to learn: \(\gamma, \beta\)

  • Output:

    \[\{ o_{i} = BN_{\gamma, \beta} ( b_{i} ) \}\]
  • Mini-batch mean:

    \[\mu_{\beta} \leftarrow \frac{1}{m}\sum_{i=1}^{m}b_{i}\]
  • Mini-batch variance:

    \[\sigma_{\beta }^{2}\leftarrow \frac{1}{m}\sum_{i=1}^{m} ( b_{i} - \mu_{\beta} )^{2}\]
  • Normalize:

    \[\hat{b_{i}} \leftarrow \frac{b_{i} - \mu_{\beta}}{\sqrt{\sigma_{\beta }^{2} + \epsilon }}\]
  • Scale and shift:

    \[o_{i} \leftarrow \gamma\hat{b_{i}} + \beta = BN_{\gamma ,\beta } ( b_{i} )\]