Specify Custom Layer Backward Function

If Deep Learning Toolbox™ does not provide the layer you require for your classification or regression problem, then you can define your own custom layer. For a list of built-in layers, see List of Deep Learning Layers.

The example Define Custom Deep Learning Layer with Learnable Parameters shows how to create a custom PreLU layer and goes through the following steps:

1. Name the layer — Give the layer a name so that you can use it in MATLAB®.

2. Declare the layer properties — Specify the properties of the layer including learnable parameters and state parameters.

3. Create a constructor function (optional) — Specify how to construct the layer and initialize its properties. If you do not specify a constructor function, then at creation, the software initializes the `Name`, `Description`, and `Type` properties with `[]` and sets the number of layer inputs and outputs to 1.

4. Create forward functions — Specify how data passes forward through the layer (forward propagation) at prediction time and at training time.

5. Create reset state function (optional) — Specify how to reset state parameters.

6. Create a backward function (optional) — Specify the derivatives of the loss with respect to the input data and the learnable parameters (backward propagation). If you do not specify a backward function, then the forward functions must support `dlarray` objects.

If the forward function only uses functions that support `dlarray` objects, then creating a backward function is optional. In this case, the software determines the derivatives automatically using automatic differentiation. For a list of functions that support `dlarray` objects, see List of Functions with dlarray Support. If you want to use functions that do not support `dlarray` objects, or want to use a specific algorithm for the backward function, then you can define a custom backward function using this example as a guide.

Create Custom Layer

The example Define Custom Deep Learning Layer with Learnable Parameters shows how to create a PReLU layer. A PReLU layer performs a threshold operation, where for each channel, any input value less than zero is multiplied by a scalar learned at training time.[1] For values less than zero, a PReLU layer applies scaling coefficients ${\alpha }_{i}$ to each channel of the input. These coefficients form a learnable parameter, which the layer learns during training.

The PReLU operation is given by

where ${x}_{i}$ is the input of the nonlinear activation f on channel i, and ${\alpha }_{i}$ is the coefficient controlling the slope of the negative part. The subscript i in ${\alpha }_{i}$ indicates that the nonlinear activation can vary on different channels.

View the layer created in the example Define Custom Deep Learning Layer with Learnable Parameters. This layer does not have a `backward` function.

```classdef preluLayer < nnet.layer.Layer % Example custom PReLU layer. properties (Learnable) % Layer learnable parameters % Scaling coefficient Alpha end methods function layer = preluLayer(numChannels,args) % layer = preluLayer(numChannels) creates a PReLU layer % with numChannels channels. % % layer = preluLayer(numChannels,Name=name) also specifies the % layer name. arguments numChannels args.Name = ""; end % Set layer name. layer.Name = name; % Set layer description. layer.Description = "PReLU with " + numChannels + " channels"; % Initialize scaling coefficient. layer.Alpha = rand([1 1 numChannels]); end function Z = predict(layer, X) % Z = predict(layer, X) forwards the input data X through the % layer and outputs the result Z. Z = max(X,0) + layer.Alpha .* min(0,X); end end end```

Note

If the layer has a custom backward function, then you can still inherit from `nnet.layer.Formattable`.

Create Backward Function

Implement the `backward` function that returns the derivatives of the loss with respect to the input data and the learnable parameters.

The `backward` function syntax depends on the type of layer.

• `dLdX = backward(layer,X,Z,dLdZ,memory)` returns the derivatives `dLdX` of the loss with respect to the layer input, where `layer` has a single input and a single output. `Z` corresponds to the forward function output and `dLdZ` corresponds to the derivative of the loss with respect to `Z`. The function input `memory` corresponds to the memory output of the forward function.

• `[dLdX,dLdW] = backward(layer,X,Z,dLdZ,memory)` also returns the derivative `dLdW` of the loss with respect to the learnable parameter, where `layer` has a single learnable parameter.

• `[dLdX,dLdSin] = backward(layer,X,Z,dLdZ,dLdSout,memory)` also returns the derivative `dLdSin` of the loss with respect to the state input using any of the previous syntaxes, where `layer` has a single state parameter and `dLdSout` corresponds to the derivative of the loss with respect to the layer state output.

• `[dLdX,dLdW,dLdSin] = backward(layer,X,Z,dLdZ,dLdSout,memory)` also returns the derivative `dLdW` of the loss with respect to the learnable parameter and returns the derivative `dLdSin` of the loss with respect to the layer state input using any of the previous syntaxes, where `layer` has a single state parameter and single learnable parameter.

You can adjust the syntaxes for layers with multiple inputs, multiple outputs, multiple learnable parameters, or multiple state parameters:

• For layers with multiple inputs, replace `X` and `dLdX` with `X1,...,XN` and `dLdX1,...,dLdXN`, respectively, where `N` is the number of inputs.

• For layers with multiple outputs, replace `Z` and `dLdZ` with `Z1,...,ZM` and `dLdZ1,...,dLdZM`, respectively, where `M` is the number of outputs.

• For layers with multiple learnable parameters, replace `dLdW` with `dLdW1,...,dLdWP`, where `P` is the number of learnable parameters.

• For layers with multiple state parameters, replace `dLdSin` and `dLdSout` with `dLdSin1,...,dLdSinK` and `dLdSout1,...,dLdSoutK`, respectively, where `K` is the number of state parameters.

To reduce memory usage by preventing unused variables being saved between the forward and backward pass, replace the corresponding input arguments with `~`.

Tip

If the number of inputs to `backward` can vary, then use `varargin` instead of the input arguments after `layer`. In this case, `varargin` is a cell array of the inputs, where the first `N` elements correspond to the `N` layer inputs, the next `M` elements correspond to the `M` layer outputs, the next `M` elements correspond to the derivatives of the loss with respect to the `M` layer outputs, the next `K` elements correspond to the `K` derivatives of the loss with respect to the `K` states outputs, and the last element corresponds to `memory`.

If the number of outputs can vary, then use `varargout` instead of the output arguments. In this case, `varargout` is a cell array of the outputs, where the first `N` elements correspond to the `N` the derivatives of the loss with respect to the `N` layer inputs, the next `P` elements correspond to the derivatives of the loss with respect to the `P` learnable parameters, and the next `K` elements correspond to the derivatives of the loss with respect to the `K` state inputs.

Note

`dlnetwork` objects do not support custom layers that require a memory value in a custom backward function. To use a custom layer with a custom backward function in a `dlnetwork` object, the `memory` input of the `backward` function definition must be `~`.

Because a PReLU layer has only one input, one output, one learnable parameter, and does not require the outputs of the layer forward function or a memory value, the syntax for `backward` for a PReLU layer is ```[dLdX,dLdAlpha] = backward(layer,X,~,dLdZ,~)```. The dimensions of `X` are the same as in the forward function. The dimensions of `dLdZ` are the same as the dimensions of the output `Z` of the forward function. The dimensions and data type of `dLdX` are the same as the dimensions and data type of `X`. The dimension and data type of `dLdAlpha` is the same as the dimension and data type of the learnable parameter `Alpha`.

During the backward pass, the layer automatically updates the learnable parameters using the corresponding derivatives.

To include a custom layer in a network, the layer forward functions must accept the outputs of the previous layer and forward propagate arrays with the size expected by the next layer. Similarly, when `backward` is specified, the `backward` function must accept inputs with the same size as the corresponding output of the forward function and backward propagate derivatives with the same size.

The derivative of the loss with respect to the input data is

`$\frac{\partial L}{\partial {x}_{i}}=\frac{\partial L}{\partial f\left({x}_{i}\right)}\frac{\partial f\left({x}_{i}\right)}{\partial {x}_{i}}$`

where $\partial L/\partial f\left({x}_{i}\right)$ is the gradient propagated from the next layer, and the derivative of the activation is

The derivative of the loss with respect to the learnable parameters is

`$\frac{\partial L}{\partial {\alpha }_{i}}=\sum _{j}^{}\frac{\partial L}{\partial f\left({x}_{ij}\right)}\frac{\partial f\left({x}_{ij}\right)}{\partial {\alpha }_{i}}$`

where i indexes the channels, j indexes the elements over height, width, and observations, and the gradient of the activation is

Create the backward function that returns these derivatives.

``` function [dLdX, dLdAlpha] = backward(layer, X, ~, dLdZ, ~) % [dLdX, dLdAlpha] = backward(layer, X, ~, dLdZ, ~) % backward propagates the derivative of the loss function % through the layer. % Inputs: % layer - Layer to backward propagate through % X - Input data % dLdZ - Gradient propagated from the deeper layer % Outputs: % dLdX - Derivative of the loss with respect to the % input data % dLdAlpha - Derivative of the loss with respect to the % learnable parameter Alpha dLdX = layer.Alpha .* dLdZ; dLdX(X>0) = dLdZ(X>0); dLdAlpha = min(0,X) .* dLdZ; dLdAlpha = sum(dLdAlpha,[1 2]); % Sum over all observations in mini-batch. dLdAlpha = sum(dLdAlpha,4); end```

Complete Layer

View the completed layer class file.

```classdef preluLayer < nnet.layer.Layer % Example custom PReLU layer. properties (Learnable) % Layer learnable parameters % Scaling coefficient Alpha end methods function layer = preluLayer(numChannels,args) % layer = preluLayer(numChannels) creates a PReLU layer % with numChannels channels. % % layer = preluLayer(numChannels,Name=name) also specifies the % layer name. arguments numChannels args.Name = ""; end % Set layer name. layer.Name = name; % Set layer description. layer.Description = "PReLU with " + numChannels + " channels"; % Initialize scaling coefficient. layer.Alpha = rand([1 1 numChannels]); end function Z = predict(layer, X) % Z = predict(layer, X) forwards the input data X through the % layer and outputs the result Z. Z = max(X,0) + layer.Alpha .* min(0,X); end function [dLdX, dLdAlpha] = backward(layer, X, ~, dLdZ, ~) % [dLdX, dLdAlpha] = backward(layer, X, ~, dLdZ, ~) % backward propagates the derivative of the loss function % through the layer. % Inputs: % layer - Layer to backward propagate through % X - Input data % dLdZ - Gradient propagated from the deeper layer % Outputs: % dLdX - Derivative of the loss with respect to the % input data % dLdAlpha - Derivative of the loss with respect to the % learnable parameter Alpha dLdX = layer.Alpha .* dLdZ; dLdX(X>0) = dLdZ(X>0); dLdAlpha = min(0,X) .* dLdZ; dLdAlpha = sum(dLdAlpha,[1 2]); % Sum over all observations in mini-batch. dLdAlpha = sum(dLdAlpha,4); end end end```

GPU Compatibility

If the layer forward functions fully support `dlarray` objects, then the layer is GPU compatible. Otherwise, to be GPU compatible, the layer functions must support inputs and return outputs of type `gpuArray` (Parallel Computing Toolbox).

Many MATLAB built-in functions support `gpuArray` (Parallel Computing Toolbox) and `dlarray` input arguments. For a list of functions that support `dlarray` objects, see List of Functions with dlarray Support. For a list of functions that execute on a GPU, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox). To use a GPU for deep learning, you must also have a supported GPU device. For information on supported devices, see GPU Support by Release (Parallel Computing Toolbox). For more information on working with GPUs in MATLAB, see GPU Computing in MATLAB (Parallel Computing Toolbox).

References

[1] "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification." In 2015 IEEE International Conference on Computer Vision (ICCV), 1026–34. Santiago, Chile: IEEE, 2015. https://doi.org/10.1109/ICCV.2015.123.