Introduction

Despite their remarkable success, Transformers face a significant challenge when dealing with long sequences due to the quadratic complexity of the attention mechanism. This limitation hinders their application to tasks involving extensive contextual information, such as processing long documents or high-resolution images. While various approaches have been proposed to address this issue, they often sacrifice accuracy, specialize in specific domains, or neglect individual token-to-token interactions. To overcome these limitations, we introduce TaylorShift, a novel method that reformulates the softmax function in the attention mechanism using the Taylor approximation of the exponential. By combining this approximation with a tensor-product-based operator, TaylorShift achieves linear-time complexity while preserving the essential token-to-token interactions. We analyze the efficiency of TaylorShift in depth, both analytically and empirically and find that it outperforms the standard transformer architecture in 4 out of 5 tasks.

How does TaylorShift work?

Essentially, TaylorShift works by replacing the exponential function in the softmax by its Taylor approximation. For a vector $\mathbf x = [x_1, …, x_m] = [x_i]_{i = 1}^m$: $$ \text{softmax}(x) = \left[\frac{\exp(x_i)}{\sum_{j} \exp(x_j)}\right]_{i = 1}^m \approx \left[ \frac{\frac{x_i^2}{2} + x_i + 1}{\sum_j \frac{x_j^2}{2} + x_j + 1} \right]_{i = 1}^m = \text{T-SM}(x) $$

Direct TaylorShift

We call the direct implementation of the attention mechanism using the Taylor Softmax direct-TaylorShift, as seen here. For queries $Q$, keys $K$, and values $V$, this becomes: $$ Y = \text{T-SM}(Q K^\top) V $$

Efficient TaylorShift

Direct-TaylorShift has the same scaling behavior as standard attention. However, we can reduce its computational complexity from $\mathcal O(N^2 d)$ to $\mathcal O(N d^3)$ by reordering the operations internally. This becomes useful for long sequences, where $N \gg d$.

Let me first introduce a tensor-product-based operator: $$ \boxtimes: \mathbb R^{N \times d} \times \mathbb R^{N \times d} \to \mathbb R^{N \times d^2}. $$ Basically, we take two lists of $d$-dimensional vectors $[a_i \in \mathbb R^d]_i$ and $[b_i \in \mathbb R^i]_i$ and for each index $i$ we multiply each element of $a_i$ with all the elements of $b_i$. The result is $d^2$ dimensional, since that is the number of possible combinations. We also write $A^{\boxtimes 2} := A \boxtimes A$.

It turns out, that by using this operator, we can calculate TaylorShift more efficiently: $$ Y = Y_\text{nom} \oslash Y_\text{denom} = \left[ \frac{[Y_\text{nom}]_{i, :}}{[Y_\text{denom}]_i} \right]_{i = 1}^N $$ with $$ Y_\text{nom} = \frac 1 2 Q^{\boxtimes 2} \left( (K^{\boxtimes 2})^\top V \right) + Q (K^\top V) + \sum_\text{columns} V. $$ $Y_\text{denom}$ is the same, but with $\mathbb 1 = [1, …, 1]$ instead of $V$.

Normalization

We found that some intermediate results of TaylorShift tended to have very large norms, which ultimately led to training failures. We introduce the following three steps for normalization:

1. Normalize the queries and keys to one and introduce an additional attention temperature parameter (per attention-head) $\tau \in \mathbb R$: $$ q_i \gets \frac{\tau q_i}{||q_i||_2}, \hspace{10pt} k_i \gets \frac{k_i}{||k_i||_2} \hspace{10pt} \forall i=1, …, N $$
2. Counteract the scaling behaviors by multiplying $Q$ and $K$ by $\sqrt[4]{d}$ and $V$ by $\frac 1 N$.
3. Normalize the output by multiplying by $\sqrt{\frac N d}$.

Efficient-TaylorShift Algorithm

We compile all the information into the pseudocode for efficient-TaylorShift:

Find the PyTorch implementation here.

How efficient is efficient-TaylorShift?

We analyze the circumstances where efficient-TaylorShift is more efficient than direct-TaylorShift in terms of speed, based on the number of floating point operations, and memory, based on the size of intermediate results.

Floating Point Operations

The number of floating point operations for direct-TaylorShift and efficient-TaylorShift is $$\text{ops}_\text{dir} = 4N^2 d + 6 N^2,$$ $$\text{ops}_\text{eff} = N (4d^3 + 10d^2 + 9d + 4).$$

Therefore, there exists an $N_0 = N_0(d)$, such that efficient-TaylorShift is more efficient for all $N > N_0$. We calculate $$ N_0 = \frac{4d^3 + 10d^2 + 9d + 4}{4d + 6} \leq d^2 + d + \frac 3 4. $$

Size of intermediate Results

The size of the largest intermediate results needed at once for direct- and efficient-TaylorShift is $$\text{entries}_\text{dir} = \underbrace{dN}_{\text{for } V} + \underbrace{2N^2}_{\text{for } QK^\top \text{ and result}},$$ $$\text{entries}_\text{eff} = d^2(d+1) + 2dN + (d+1)N + d^2N.$$

We can again find $N_1 = N_1(d)$, such that efficient-TaylorShift is more memory efficient for all $N > N_1$. We find $$ N_1 = \frac 1 4 \left[ d^2 + 2 d + 1 + \sqrt{d^4 + 12 d^3 + 14 d^2 + 4d + 1} \right] \leq \frac 1 2 d^2 + 2 d + \frac 1 2. $$

How can we increase the efficiency?

In an effort to increase the efficiency while processing the same number of tokens $N$, one might opt to reduce the embedding dimension $d_\text{emb}$. However, this might come at the cost of expressiveness. Given that efficient-TaylorShift scales with $\mathcal O(Nd^3)$, we can instead increase the number of attention heads $h$ with dimension $d = \frac{d_\text{emb}}{h}$ each. We find that the number of operations is $$ \text{ops}_\text{eff}(\text{MHSA}) = N \left( 4 \frac{d_\text{emb}^3}{h^2} + 10 \frac{d_\text{emb}^2}{h} + 9 d_\text{emb} + 4h \right) $$ and the number of entries is $$ \text{entries}_\text{eff}(\text{MHSA}) = \frac{d_\text{emb}^3}{h^2} + (N + 1) \frac{d_\text{emb}^2}{h} + 3N d_\text{emb} + N h, $$ which are both strictly decreasing in $h$. Therefore, efficient-TaylorShift becomes faster and needs less memory with more attention heads.

Empirical Evaluation

Efficiency of TaylorShift Attention

We first validate our theoretical analysis on the efficiency of TaylorShift by measuring its inference time and memory usage under different configurations of $d$ and $N$:

Performance of a Transformer with TaylorShift

We test the accuracy of multiple (efficient) Transformers on a set of 5 tasks from the Long Range Arena benchmark [4], as well as ImageNet classification at two model sizes. We use the same standard hyperparameters for all models. Models with * had to be trained in full instead of mixed precision. All tasks are classitication tasks and the table shows accuracy in percent.

This shows TaylorShift’s consistent performance across various datasets. It surpasses all other models on 4 out of 5 datasets, positioning itself as a good all-rounder model. We observe a notable increase of $4.3\%$ when transitioning from size Ti to S on ImageNet, in contrast to $3.5\%$ for the Transformer, which highlights TaylorShifts scalability.

Number of attention heads

We train TaylorShift models on the pixel-level CIFAR10 task to see how accuracy and efficiency change. All models have the default $d_\text{emb} = 256$ with $d = \frac{d_\text{emb}}{h}$ in the attention mechanism. The default is $h = 4$.

We see that increasing the number of attention heads $h$ increases the speed and decreases the memory needed by efficient-TaylorShift, as predicted. Additionally, we find that it also increases the performance up to a certain point. Until there, we have a win-win-win situation with a faster, more lightweight and more accurate model. After that the number of heads can be used to trade off accuracy against the amount compute needed.

Conclusion & Outlook

We introduced TaylorShift a novel efficient Transformer model. It offers significant computational advantages without sacrificing performance. By approximating the exponential function, TaylorShift achieves linear time and memory complexity, making it ideal for long sequences. Our experiments demonstrate its superiority over standard Transformers in terms of speed, memory efficiency, and even accuracy.

As we move forward, we envision TaylorShift opening up new possibilities for tackling challenging tasks involving lengthy sequences. From high-resolution image processing to large-scale document analysis, TaylorShift’s efficiency and versatility make it a promising tool for the future of efficient Transformer models.

References

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Citation

If you use this information, method or the associated code, please cite our paper:

@misc{Nauen2024TaylorShift,
      title         = {TaylorShift: Shifting the Complexity of Self-Attention from Squared to Linear (and Back) using Taylor-Softmax},
      author        = {Tobias Christian Nauen and Sebastian Palacio and Andreas Dengel},
      year          = {2024},
      eprint        = {2403.02920},
      archivePrefix = {arXiv},
      primaryClass  = {cs.LG},
      note          = {Accepted at ICPR 2024},
}