Sequential

Composes an array of functions into a new function that calls these functions one after another ("forward function composition").

Sequential (arrayOfFunctions)

Parameters

arrayOfFunctions: a BrainScript array of functions, e.g. constructed with the : operator: (LinearLayer{1024} : Sigmoid)

Return value

This function returns another function. That returned function takes one argument, and returns the result of applying all given functions in sequence to the input.

Description

Sequential() is a powerful operation that allows to compactly express a very common situation in neural networks where an input is processed by propagating it through a progression of layers. You may be familiar with it from other neural-network toolkits.

Sequential() takes an array of functions as its argument, and returns a new function that invokes these function in order, each time passing the output of one to the next. Consider this example:

FGH = Sequential (F:G:H)
y = FGH (x)

Here, the colon (:) is BrainScript's syntax of expressing arrays. For example, (F:G:H) is an array with three elements, F, G, and H. In Python, for example, this would be written as [ F, G, h ].

The FGH function defined above means the same as

y = H(G(F(x))) 

This is known as "function composition", and is especially convenient for expressing neural networks, which often have this form:

     +-------+   +-------+   +-------+
x -->|   F   |-->|   G   |-->|   H   |--> y
     +-------+   +-------+   +-------+

Lastly, please be aware that the following expression:

z = Sequential (DenseLayer{1024} : DenseLayer{1024}) (x)

means something different from:

layer = DenseLayer{1024}
z = Sequential (layer : layer) (x)

In the latter form, the same function with the same shared set of parameters is applied twice, while in the former, the two layers have separate sets of parameters.

Example

Standard 4-hidden layer feed-forward network as used in the earlier deep-neural network work on speech recognition:

model = Sequential (
    DenseLayer{2048, activation=Sigmoid} :  # four hidden layers
    DenseLayer{2048, activation=Sigmoid} : 
    DenseLayer{2048, activation=Sigmoid} : 
    DenseLayer{2048, activation=Sigmoid} : 
    DenseLayer{9000, activation=Softmax}    # last is a Softmax 
)
features = Input{40}
p = model (features)