Backpropogating an LSTM: A Numerical Example

Let’s do this…

We all know LSTM’s are super powerful; So, we should know how they work and how to use them.

Syntactic notes

Above is the element-wise product or Hadamard product.

Inner products will be represented as

Outer products will be respresented as

represents the sigmoid function:

The forward components

The gates are defined as:

Input activation:

Input gate:

Forget gate:

Output gate:



Note for simplicity we define:

Which leads to:

Internal state:

Output:



The backward components

Given:

the output difference as computed by any subsequent layers (i.e. the rest of your network), and;

the output difference as computed by the next time-step LSTM (the equation for t-1 is below).

Find:

The final updates to the internal parameters is computed as:

Putting this all together we can begin…

The Example

Let us begin by defining out internal weights:

And now input data:

I’m using a sequence length of two here to demonstrate the unrolling over time of RNNs

Forward @

From here, we can pass forward our state and output and begin the next time-step.

Forward @

And since we’re done our sequence we have everything we need to begin backpropogating.

Backward @

First we’ll need to compute the difference in output from the expected (label).

Note for this we’ll be using L2 Loss: . The derivate w.r.t. is .

because there are no future time-steps.

Now we can pass back our and continue on computing…

Backward @

And we’re done the backward step!

Now we’ll need to update our internal parameters according to whatever solving algorithm you’ve chosen. I’m going to use a simple Stochastic Gradient Descent (SGD) update with learning rate: .

We’ll need to compute how much our weights are going to change by:

And updating out parameters based on the SGD update function: we get our new weight set:

And that completes one iteration of solving an LSTM cell!

Of course, this whole process is sequential in nature and a small error will render all subsequent calculations useless, so if you catch ANYTHING email me at hello@aidangomez.ca