BatchNormInference

Versioned name: BatchNormInference-5

Category: Normalization

Short description: BatchNormInference performs Batch Normalization operation described in the Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift article.

Detailed Description

BatchNormInference performs the following operations on a given data batch input tensor data:

• Normalizes each activation \(x^{(k)}\) by the mean and variance.

  \[\hat{x}^{(k)}=\frac{x^{(k)} - E[x^{(k)}]}{\sqrt{Var(x^{(k)}) + \epsilon}}\]

  where \(E[x^{(k)}]\) and \(Var(x^{(k)})\) are the mean and variance, calculated per channel axis of data input, and correspond to mean and variance inputs, respectively. Additionally, \(\epsilon\) is a value added to the variance for numerical stability and corresponds to epsilon attribute.

• Performs linear transformation of each normalized activation based on gamma and beta input, representing the scaling factor and shift, respectively.

  \[\hat{y}^{(k)}=\gamma^{(k)}\hat{x}^{(k)} + \beta^{(k)}\]

  where \(\gamma^{(k)}\) and \(\beta^{(k)}\) are learnable parameters, calculated per channel axis, and correspond to gamma and beta inputs.

Mathematical Formulation

Let x be a d-dimensional input, \(x=(x_{1}\dotsc x_{d})\). Since normalization is applied to each activation \(E[x^{(k)}]\), you can focus on a particular activation and omit k.

For a particular activation, consider a mini-batch \(\mathcal{B}\) of m values. BatchNormInference performs Batch Normalization algorithm as follows:

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

  \[\mathcal{B} = {x_{1...m}}\]

• Parameters to learn: \(\gamma, \beta\)

• Output:

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

• Mini-batch mean:

  \[\mu_{\mathcal{B}} \leftarrow \frac{1}{m}\sum_{i=1}^{m}b_{i}\]

• Mini-batch variance:

  \[\sigma_{\mathcal{B}}^{2}\leftarrow \frac{1}{m}\sum_{i=1}^{m} ( b_{i} - \mu_{\mathcal{B}})^{2}\]

• Normalize:

  \[\hat{b_{i}} \leftarrow \frac{b_{i} - \mu_{\mathcal{B}}}{\sqrt{\sigma_{\mathcal{B}}^{2} + \epsilon }}\]

• Scale and shift:

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

Attributes:

• epsilon

  • Description: epsilon is a constant added to the variance for numerical stability.

  • Range of values: a floating-point number greater than or equal to zero

  • Type: float

  • Required: yes

Inputs

• 1: data - A tensor of type T and at least rank 2. The second dimension represents the channel axis and must have a span of at least 1. Required.

• 2: gamma - Scaling factor for normalized value. A 1D tensor of type T with the same span as data channel axis. Required.

• 3: beta - Bias added to the scaled normalized value. A 1D tensor of type T with the same span as data channel axis. Required.

• 4: mean - Value for mean normalization. A 1D tensor of type T with the same span as data channel axis. Required.

• 5: variance - Value for variance normalization. A 1D tensor of type T with the same span as data channel axis. Required.

Outputs

• 1: The result of element-wise Batch Normalization operation applied to the input tensor data. A tensor of type T and the same shape as data input tensor.

Types

• T: any supported floating-point type.

Examples

Example: 2D input tensor data

<layer ... type="BatchNormInference" ...>
    <data epsilon="9.99e-06" />
    <input>
        <port id="0">  <!-- input -->
            <dim>10</dim>
            <dim>128</dim>
        </port>
        <port id="1">  <!-- gamma -->
            <dim>128</dim>
        </port>
        <port id="2">  <!-- beta -->
            <dim>128</dim>
        </port>
        <port id="3">  <!-- mean -->
            <dim>128</dim>
        </port>
        <port id="4">  <!-- variance -->
            <dim>128</dim>
        </port>
    </input>
    <output>
        <port id="5">
            <dim>10</dim>
            <dim>128</dim>
        </port>
    </output>
</layer>

Example: 4D input tensor data

<layer ... type="BatchNormInference" ...>
    <data epsilon="9.99e-06" />
    <input>
        <port id="0">  <!-- input -->
            <dim>1</dim>
            <dim>3</dim>
            <dim>224</dim>
            <dim>224</dim>
        </port>
        <port id="1">  <!-- gamma -->
            <dim>3</dim>
        </port>
        <port id="2">  <!-- beta -->
            <dim>3</dim>
        </port>
        <port id="3">  <!-- mean -->
            <dim>3</dim>
        </port>
        <port id="4">  <!-- variance -->
            <dim>3</dim>
        </port>
    </input>
    <output>
        <port id="5">
            <dim>1</dim>
            <dim>3</dim>
            <dim>224</dim>
            <dim>224</dim>
        </port>
    </output>
</layer>