An In-Depth Analysis of LSTM Forget Gate Activation Functions

LSTMs (Long Short-Term Memory networks) are a type of recurrent neural network (RNN) designed to address the vanishing gradient problem. A key component of an LSTM is the forget gate, which plays a crucial role in determining which parts of the cell state to retain or allow the model to forget. The forget gate uses the sigmoid activation function, which allows for a smooth, continuous transition between 0 and 1. This is in contrast to the softmax activation function, which is often misunderstood to produce a binary output of either 0 or 1. However, this is not entirely accurate.

Understanding the Sigmoid Activation Function

The sigmoid activation function is a commonly used non-linear function that transforms a real-valued number into a value between 0 and 1. Its equation is defined as:

g(x)  1 / (1   exp(-x))

The sigmoid function ensures that values are adjusted smoothly, making it ideal for the forget gate where gradual changes in the cell state are necessary.

The Softmax Activation Function: A Misconception

Softmax is not concretely 0 or 1, but a probability distribution. The softmax function is often used in classification tasks where it normalizes a vector of values into a probability distribution. Its equation is given by:

u03B8_i  exp(x_i) / sum; exp(x_j)

While the softmax function can indeed produce values close to 0 or 1, it is not inherently binary. Instead, it transforms a vector of raw scores into a vector of probabilities that sum up to 1.

Why the Sigmoid Activation is Preferred for LSTM Forget Gates

The main reason for using the sigmoid activation function for the forget gate in an LSTM is its continuous nature. This allows the model to gradually adjust the cell state based on the input and previous state, avoiding the issue of abrupt transitions.

Let's delve into the mathematical representation of the forget gate:

u03C0_f(t)  sigmoid(W_f u03C7(t-1)   U_f x(t)   b_f)

Here, u03C0_f(t) represents the forget gate output, W_f and U_f are weight matrices, x(t) is the input vector, and b_f is the bias term. The sigmoid function ensures that the output is a value between 0 and 1, allowing for fine-grained control over what to forget.

Comparison with Softmax Activation

While the softmax function can produce binary-like outputs, it is not as natural for tasks where gradual changes are needed. For instance, in the context of the LSTM forget gate, we are interested in a smooth transition that reflects the model's decision to retain or discard information. The softmax function, which outputs probabilities that sum to 1, would not be as appropriate for this task.

Conclusion

In summary, the sigmoid activation function is preferred for the forget gate in LSTM networks due to its continuous and smooth nature, which ensures gradual transitions between states. The misconception that softmax produces a binary output of either 0 or 1 is a common misunderstanding. In reality, the softmax function is more suited for classification tasks where probabilities are needed.

References

For a deeper understanding of the concepts discussed, refer to the following resources:

Rumelhart, D. E., Hinton, G. E., Williams, R. J. (1986). Learning internal representations by error propagation. In D. E. Rumelhart, J. L. McClelland, P. R. Orfield (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition (pp. 318-362). MIT Press. Sutskever, I., Vinyals, O., Le, Q. V. (2014). Sequence to sequence learning with neural networks. In Advances in neural information processing systems (pp. 3104-3112).