The standard self-attention mechanism, while powerful, carries a significant computational burden, especially as input sequences grow longer. Understanding this computational cost is fundamental to appreciating the motivation behind more efficient architectures.
Let's analyze the core operations within the scaled dot-product attention formula, which forms the basis of self-attention in Transformers:
Here, (Query), (Key), and (Value) are matrices derived from the input sequence embeddings. Let be the sequence length and be the dimension of the keys and queries, and be the dimension of the values.
The primary computational steps are:
Query-Key Similarity Calculation: Computing the matrix product .
Scaling and Softmax: Scaling the scores by and applying the softmax function row-wise.
Value Aggregation: Multiplying the softmax output (the attention weights matrix , dimensions ) by the Value matrix .
Combining these steps, the total computational complexity is dominated by the two large matrix multiplications: .
In many standard Transformer configurations, the dimensions and are proportional to the overall model embedding dimension (often , where is the number of attention heads). Therefore, the complexity is commonly summarized as:
This quadratic dependence on the sequence length is the critical bottleneck. While the operations are highly parallelizable across the sequence dimension (unlike recurrent models), the total number of operations grows rapidly with .
Beyond computation, there's also a significant memory requirement. The intermediate attention score matrix (before or after softmax) has dimensions . Storing this matrix requires:
memory. For long sequences (e.g., ), storing an matrix of floating-point numbers can exceed the memory capacity of typical accelerators like GPUs, even before considering the memory needed for activations, gradients, and model parameters.
The graph illustrates the rapid divergence in computational cost. As sequence length increases, the cost of standard self-attention quickly surpasses linear growth, making it impractical for very long sequences. The Y-axis uses a logarithmic scale to accommodate the large range of values.
This quadratic computational and memory complexity severely limits the application of standard Transformers to tasks involving very long sequences, such as:
Even with powerful hardware, sequence lengths past a few thousand tokens become challenging from both a runtime and memory perspective during training and inference. This limitation directly motivates the development of the alternative attention mechanisms and architectural variants discussed in the following sections, which aim to reduce this quadratic dependency.
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