Computational Efficiency Considerations

One of the motivations behind the development of Gated Recurrent Units (GRUs) was to create a gated recurrent unit that retains the ability to manage long-range dependencies, similar to LSTMs, but with a simpler structure and consequently, greater computational efficiency. This efficiency stems primarily from the GRU's reduced complexity compared to the LSTM cell. The factors contributing to this difference are analyzed.

Fewer Parameters

The most direct contributor to GRU's efficiency is its lower parameter count compared to an LSTM cell with the same hidden state size. Recall the structures:

• LSTM: Has three distinct gates (forget $f_t$, input $i_t$, output $o_t$) and a candidate cell state calculation ($\tilde{C}_t$). Each of these typically involves a separate weight matrix and bias vector for the input $x_t$ and the previous hidden state $h_{t-1}$. This leads to four sets of parameters for these transformations.
• GRU: Has only two gates (reset $r_t$, update $z_t$). The candidate hidden state $\tilde{h}_t$ calculation uses the reset gate, and the final hidden state $h_t$ is computed using the update gate. This structure requires only three sets of parameters for these transformations.

Let $d$ be the input feature dimension and $h$ be the hidden state dimension.

For an LSTM cell, the approximate number of parameters (weights and biases) is: $\text{Params}_{\text{LSTM}} \approx 4 \times (h \times (d + h) + h)$ The '4' comes from the four transformations (input gate, forget gate, output gate, candidate cell state). Each transformation involves weights mapping the concatenated input $x_t$ (dimension $d$) and previous hidden state $h_{t-1}$ (dimension $h$) to the hidden dimension $h$, plus a bias vector of dimension $h$.

For a GRU cell, the approximate number of parameters is: $\text{Params}_{\text{GRU}} \approx 3 \times (h \times (d + h) + h)$ The '3' comes from the three transformations (reset gate, update gate, candidate hidden state). The structure is similar, but with one fewer gate-related transformation compared to LSTM.

This means a GRU cell typically has about 25% fewer parameters than an LSTM cell with the same hidden state size.

Comparison of the approximate number of parameter sets involved in the core transformations within LSTM and GRU cells.

Reduced Computational Load

Fewer parameters directly translate to fewer computations required per time step. The core operations in both LSTMs and GRUs involve matrix multiplications (between inputs/hidden states and weight matrices) and element-wise operations (for gate activations and state updates).

Since a GRU performs three main matrix multiplication steps per time step compared to LSTM's four, it requires fewer floating-point operations (FLOPs). This reduction applies to both the forward pass (calculating hidden states) and the backward pass (calculating gradients during training via Backpropagation Through Time).

While the exact speedup depends on the specific hardware, software implementations (like cuDNN optimizations), and model dimensions, GRUs generally execute faster per time step.

Impact on Training and Inference

• Training Speed: With fewer computations per forward and backward pass, training GRU networks can be faster than training equivalent LSTM networks, especially for large datasets or complex models where the computational cost of the recurrent layers is significant. This means you might achieve convergence in less wall-clock time or be able to run more training epochs within a given time budget.
• Inference Speed: The reduced computational load is also beneficial during inference (when using the trained model to make predictions). Faster predictions are often desirable, especially in real-time applications like live translation or responsive user interfaces.
• Memory Usage: A model with fewer parameters consumes less memory. This is advantageous both during training (potentially allowing for larger batch sizes or models on memory-constrained GPUs) and deployment, particularly on edge devices or mobile phones where memory resources are often limited.

Efficiency vs. Performance

It is important to remember that computational efficiency is only one aspect of model selection. While GRUs are often faster and lighter than LSTMs, LSTMs, with their distinct cell state and output gate, sometimes offer slightly better performance on tasks requiring the modeling of particularly intricate or long-range dependencies. The choice between GRU and LSTM often involves an empirical evaluation, balancing the need for computational resources against the desired predictive performance, as we will discuss further in the next section.