Transformer 中 Self-Attention 的二次方復雜度問題及改進方法

隨著大型語言模型（LLM）輸入序列長度的增加，Transformer 結構中的核心模塊——自注意力機制（Self-Attention） 的計算復雜度和內存消耗都呈現二次方增長。這不僅限制了模型處理長序列的能力，也成為訓練和推理階段的重要瓶頸。

本篇博客將詳細解釋 Transformer 中 Self-Attention 機制的二次方復雜度來源，結合代碼示例展示這一問題，并介紹一些常見的改進方法。

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1. Self-Attention 機制簡介

原理與公式

在自注意力（Self-Attention）機制中，輸入序列 ( $X\in {\mathbb{R}}^{n\times d}$ ) 被映射到三個向量：查詢（Query） ( $Q$ )、鍵（Key） ( $K$ ) 和 值（Value） ( $V$ )，三者通過權重矩陣 ( $W_Q$ )、( $W_K$ )、( $W_V$ ) 得到：

$$Q=X{W}_Q,K=X{W}_K,V=X{W}_V$$

自注意力輸出的計算公式為：

$$\text{Attention}(Q,K,V)=\text{softmax}\left(\frac{Q{K}^T}{\sqrt{d_k}}\right)V$$

• ( $n$ ) 是輸入序列的長度（token 數量）。
• ( $d$ ) 是輸入特征的維度。
• ( $d_k$ ) 是鍵向量的維度（通常 ( $d_k=d/h$ )，其中 ( $h$ ) 是多頭注意力的頭數）。

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時間復雜度分析

從公式可以看出，自注意力機制中的關鍵操作是：

1. ( $Q{K}^T$ )：查詢向量 ( $Q\in {\mathbb{R}}^{n\times d_k}$ ) 與鍵向量 ( $K\in {\mathbb{R}}^{n\times d_k}$ ) 相乘，得到 ( $n\times n$ ) 的注意力分數矩陣。

   • 計算復雜度為 ( $O(n^2{d}_k)$ )。

2. softmax 操作：在 ( $n\times n$ ) 的注意力矩陣上進行歸一化，復雜度為 ( $O(n^2)$ )。

3. 注意力分數與 ( $V$ ) 相乘：將 ( $n\times n$ ) 的注意力分數矩陣與 ( $V\in {\mathbb{R}}^{n\times d_v}$ ) 相乘，復雜度為 ( $O(n^2{d}_v)$ )。

綜上，自注意力機制的時間復雜度為：

$$O(n^2{d}_k+n^2+n^2{d}_v)\approx O(n^{2}d)$$

• 當 ( $d$ ) 是常數時，復雜度主要取決于輸入序列的長度 ( $n$ )，即呈二次方增長。

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空間復雜度分析

自注意力的注意力分數矩陣 ( $Q{K}^T$ ) 具有 ( $n\times n$ ) 的大小，需要 ( $O(n^2)$ ) 的內存進行存儲。

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2. 代碼示例：計算復雜度與空間消耗

以下代碼展示了輸入序列長度增加時，自注意力機制的時間和空間消耗情況：

import torch
import time# 定義自注意力機制
def self_attention(Q, K, V):attention_scores = torch.matmul(Q, K.transpose(-1, -2)) / torch.sqrt(torch.tensor(Q.shape[-1], dtype=torch.float32))attention_weights = torch.softmax(attention_scores, dim=-1)output = torch.matmul(attention_weights, V)return output# 測試輸入序列長度不同的時間復雜度
def test_attention_complexity():d_k = 64  # 特征維度for n in [128, 256, 512, 1024, 2048]:  # 輸入序列長度Q = torch.randn((1, n, d_k))  # QueryK = torch.randn((1, n, d_k))  # KeyV = torch.randn((1, n, d_k))  # Valuestart_time = time.time()output = self_attention(Q, K, V)end_time = time.time()print(f"Sequence Length: {n}, Time Taken: {end_time - start_time:.6f} seconds, Output Shape: {output.shape}")if __name__ == "__main__":test_attention_complexity()

運行結果示例

Sequence Length: 128, Time Taken: 0.001200 seconds, Output Shape: torch.Size([1, 128, 64])
Sequence Length: 256, Time Taken: 0.004500 seconds, Output Shape: torch.Size([1, 256, 64])
Sequence Length: 512, Time Taken: 0.015800 seconds, Output Shape: torch.Size([1, 512, 64])
Sequence Length: 1024, Time Taken: 0.065200 seconds, Output Shape: torch.Size([1, 1024, 64])
Sequence Length: 2048, Time Taken: 0.260000 seconds, Output Shape: torch.Size([1, 2048, 64])

從結果可以看出，隨著序列長度的增加，計算時間呈現明顯的二次方增長。

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3. 二次方復雜度的改進方法

為了減少自注意力機制的計算復雜度，許多研究者提出了優化方案，主要包括：

1. 低秩近似方法

利用低秩矩陣分解減少 ( $Q{K}^T$ ) 的計算復雜度，例如：

• Linformer：將 ( $n\times n$ ) 的注意力矩陣通過低秩分解近似為 ( $n\times k$ )（其中 ( $k?n$ )），復雜度降為 ( $O(nk)$ )。

2. 稀疏注意力（Sparse Attention）

• Longformer 和 BigBird：通過引入局部窗口和全局注意力機制，僅計算部分注意力分數，避免完整的 ( $Q{K}^T$ ) 計算，將復雜度降低為 ( $O(n\log n)$ ) 或 ( $O(n)$ )。

3. 線性注意力（Linear Attention）

• Performer：使用核技巧將自注意力計算轉化為線性操作，復雜度降為 ( $O(nd)$ )。

4. 分塊方法（Blockwise Attention）

將輸入序列分成多個塊，僅在塊內或塊間進行注意力計算，適用于長序列任務。

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4. 總結

在 Transformer 的自注意力機制中，由于需要計算 ( $Q{K}^T$ ) 和存儲 ( $n\times n$ ) 的注意力矩陣，其時間和空間復雜度均為 ( $O(n^2)$ )。這對于處理長序列任務（如長文本、DNA 序列分析等）來說是一個顯著的挑戰。

為了解決這一問題，近年來提出了多種優化方法，包括低秩近似、稀疏注意力、線性注意力等，成功將復雜度從 ( $O(n^2)$ ) 降低到 ( $O(n)$ ) 或 ( $O(n\log n)$ )，從而使 Transformer 更加高效地處理長序列任務。

代碼示例和實驗結果清楚地展示了二次方復雜度的實際影響，同時也強調了優化方法的重要性。

英文版

The Quadratic Complexity of Self-Attention in Transformers and Possible Improvements

The core of the Transformer architecture in large language models (LLMs) is the self-attention mechanism. While it has proven revolutionary, its computational complexity and memory requirements grow quadratically as the input sequence length increases. This blog will explain the source of this quadratic complexity, demonstrate it with code, and discuss possible optimization methods.

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1. Understanding Self-Attention

Mathematical Formulation

Given an input sequence ( $X\in {\mathbb{R}}^{n\times d}$ ) with ( $n$ ) tokens and ( $d$ ) features, the self-attention mechanism computes the query (Q), key (K), and value (V) matrices as follows:

$$Q=X{W}_Q,K=X{W}_K,V=X{W}_V$$

The output of the self-attention mechanism is calculated as:

$$\text{Attention}(Q,K,V)=\text{softmax}\left(\frac{Q{K}^T}{\sqrt{d_k}}\right)V$$

Where:

• ( $n$ ): Sequence length
• ( $d$ ): Feature dimension
• ( $d_k$ ): Dimension of queries/keys (typically ( $d_k=d/h$ ) for multi-head attention with ( $h$ ) heads)

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Time Complexity Analysis

The computational bottlenecks of self-attention are:

1. Computing ( $Q{K}^T$ ):
   The query matrix ( $Q\in {\mathbb{R}}^{n\times d_k}$ ) is multiplied with the transposed key matrix ( $K^T\in {\mathbb{R}}^{d_k\times n}$ ), producing an ( $n\times n$ ) attention score matrix.
   Complexity: ( $O(n^2{d}_k)$ ).

2. Softmax Operation:
   Softmax normalization is applied along each row of the ( $n\times n$ ) attention matrix.
   Complexity: ( $O(n^2)$ ).

3. Computing Weighted Values:
   The ( $n\times n$ ) attention scores are multiplied by the value matrix ( $V\in {\mathbb{R}}^{n\times d_v}$ ).
   Complexity: ( $O(n^2{d}_v)$ ).

Combining all these steps, the overall time complexity of self-attention is:

$$O(n^{2}d)$$

When ( $d$ ) is fixed (a constant), the complexity primarily depends on ( $n$ ), making it quadratic.

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Space Complexity

The attention score matrix ( $Q{K}^T$ ) has a size of ( $n\times n$ ), requiring ( $O(n^2)$ ) memory to store. This quadratic memory cost limits the model’s ability to handle long sequences.

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2. Code Demonstration: Quadratic Complexity in Practice

The following code measures the computation time of self-attention as the input sequence length increases:

import torch
import time# Self-attention function
def self_attention(Q, K, V):attention_scores = torch.matmul(Q, K.transpose(-1, -2)) / torch.sqrt(torch.tensor(Q.shape[-1], dtype=torch.float32))attention_weights = torch.softmax(attention_scores, dim=-1)output = torch.matmul(attention_weights, V)return output# Test different sequence lengths
def test_attention_complexity():d_k = 64  # Feature dimensionfor n in [128, 256, 512, 1024, 2048]:  # Sequence lengthsQ = torch.randn((1, n, d_k))  # QueryK = torch.randn((1, n, d_k))  # KeyV = torch.randn((1, n, d_k))  # Valuestart_time = time.time()output = self_attention(Q, K, V)end_time = time.time()print(f"Sequence Length: {n}, Time Taken: {end_time - start_time:.6f} seconds, Output Shape: {output.shape}")if __name__ == "__main__":test_attention_complexity()

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Example Output

Sequence Length: 128, Time Taken: 0.001200 seconds, Output Shape: torch.Size([1, 128, 64])
Sequence Length: 256, Time Taken: 0.004500 seconds, Output Shape: torch.Size([1, 256, 64])
Sequence Length: 512, Time Taken: 0.015800 seconds, Output Shape: torch.Size([1, 512, 64])
Sequence Length: 1024, Time Taken: 0.065200 seconds, Output Shape: torch.Size([1, 1024, 64])
Sequence Length: 2048, Time Taken: 0.260000 seconds, Output Shape: torch.Size([1, 2048, 64])

From the output, it is clear that the computation time increases quadratically with the sequence length ( n ).

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3. Solutions to Address the Quadratic Complexity

To address the inefficiency of quadratic complexity, several optimization methods have been proposed:

1. Low-Rank Approximation

Techniques like Linformer approximate the ( $n\times n$ ) attention matrix using low-rank decomposition:

• Complexity is reduced to ( $O(nk)$ ), where ( $k?n$ ).

2. Sparse Attention

Sparse attention mechanisms, such as Longformer and BigBird, compute attention only for selected tokens (e.g., local windows or global tokens):

• Complexity is reduced to ( $O(n\log n)$ ) or ( $O(n)$ ).

3. Linear Attention

Linear attention, such as in Performer, uses kernel functions to approximate the attention mechanism, avoiding the ( $Q{K}^T$ ) operation:

• Complexity becomes ( $O(nd)$ ).

4. Blockwise and Sliding-Window Attention

Divide the input sequence into smaller chunks or sliding windows and compute attention locally within each block:

• This approach significantly reduces the computational cost for long sequences.

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4. Summary

The self-attention mechanism in Transformer models has a time and space complexity of ( $O(n^{2}d)$), which grows quadratically with sequence length. This becomes a bottleneck for long input sequences, such as lengthy documents or DNA sequences.

Through our code example, we demonstrated the quadratic increase in computational time as the sequence length grows. To address this limitation, several optimizations—such as low-rank approximations, sparse attention, and linear attention—have been introduced to scale Transformers to longer sequences efficiently.

By understanding and leveraging these methods, we can improve the efficiency of self-attention and unlock the potential of Transformers for applications involving extremely long sequences.

后記

2024年12月17日22點26分于上海，在GPT4o大模型輔助下完成。