The Output Gate

The Long Short-Term Memory (LSTM) cell manages its internal state ($C_t$) through mechanisms like the forget gate, which discards irrelevant information, and the input gate, which incorporates new information. The output gate is the final component of the LSTM cell. This gate determines what parts of the updated cell state should be passed on as the hidden state ($h_t$) to the next time step and, potentially, as the output of the network at the current time step.

Think of the cell state ($C_t$) as the LSTM's internal, long-term memory. The output gate acts as a filter, deciding which aspects of this memory are relevant for the immediate next step and the current output. Not everything stored in the cell state might be needed right away, and the output gate allows the LSTM to selectively expose only the pertinent parts.

Filtering the Cell State

The output gate operation involves two main steps:

1. Deciding which parts of the cell state to output: Similar to the other gates, this involves a sigmoid layer. It takes the previous hidden state ($h_{t-1}$) and the current input ($x_t$) and outputs values between 0 and 1 for each number in the cell state. A value close to 1 means "let this part through," while a value close to 0 means "hold this part back." This decision vector is often denoted as $o_t$.

   The calculation is:

   $$o_t = \sigma(W_o [h_{t-1}, x_t] + b_o)$$

   Here, $\sigma$ is the sigmoid activation function, $W_o$ represents the weight matrix, and $b_o$ is the bias vector for the output gate. The term $[h_{t-1}, x_t]$ indicates that the previous hidden state and the current input vector are concatenated before being multiplied by the weights.

2. Generating the hidden state: The updated cell state ($C_t$, resulting from the forget and input gate operations) is first pushed through the hyperbolic tangent ($\tanh$) activation function. This squashes the values to be between -1 and 1, helping to regulate the numerical range of the network's signals. Then, this transformed cell state is multiplied element-wise ($\odot$) by the output gate's activation vector ($o_t$). This multiplication acts as the filter: parts of the $\tanh(C_t)$ corresponding to near-zero values in $o_t$ are diminished, while parts corresponding to near-one values in $o_t$ are passed through largely unchanged. The result of this filtering is the new hidden state, $h_t$.

   The calculation is:

   $$h_t = o_t \odot \tanh(C_t)$$

This resulting $h_t$ serves two purposes: it is the output for the current time step (if the layer is configured to return sequences or if it's the final output layer), and it becomes the $h_{t-1}$ for the next time step in the sequence.

Visualizing the Output Gate Flow

The following diagram illustrates the data flow specifically within the output gate mechanism and how it produces the hidden state $h_t$:

Data flow for calculating the output gate activation ($o_t$) and the final hidden state ($h_t$) using the current input ($x_t$), previous hidden state ($h_{t-1}$), and the updated cell state ($C_t$).

Why This Matters

The output gate is a significant part of what makes LSTMs effective. By filtering the cell state before producing the hidden state, the LSTM can maintain a rich internal representation ($C_t$) containing information potentially relevant over many time steps, while exposing only the necessary information ($h_t$) for the immediate context or task. This controlled exposure helps prevent the internal state from directly causing issues in subsequent calculations if parts of it are not immediately relevant, contributing to more stable gradients and better learning of long-range dependencies compared to simple RNNs.

In summary, the output gate completes the LSTM cycle by:

1. Using a sigmoid function to determine which parts of the cell state are relevant for the output ($o_t$).
2. Applying a $\tanh$ function to the cell state to scale it.
3. Multiplying the scaled cell state by the output gate's filter ($o_t$) to produce the hidden state ($h_t$).

This hidden state $h_t$ then carries the filtered, relevant information forward to the next time step, influencing future calculations and potentially serving as the network's output for the current step. Understanding these three gates (forget, input, output) and the cell state is fundamental to grasping the power and utility of LSTM networks in sequence modeling.