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Linear Log-Normal Attention with Unbiased Concentration

Yury Nahshan, Joseph Kampeas, Emir Haleva

TL;DR

Addressing the quadratic $O(N^2)$ cost of self-attention, the paper analyzes the distribution and concentration of the SA matrix and introduces Linear Log-Normal Attention (LLN), a linear-time mechanism built from exponential feature maps to mimic SA's log-normal distribution. LLN employs moment matching to align its variance with SA and uses a temperature parameter to control concentration, supported by entropy and spectral-gap analyses that distinguish unbiased from biased concentration. Empirically, LLN matches SA performance on NLP benchmarks while offering substantial scalability, and a block-diagonal augmentation further strengthens short-range interactions. Overall, the work provides a rigorous framework for scalable attention by coupling distributional mimicry with concentration metrics, and it releases code to enable broader use.

Abstract

Transformer models have achieved remarkable results in a wide range of applications. However, their scalability is hampered by the quadratic time and memory complexity of the self-attention mechanism concerning the sequence length. This limitation poses a substantial obstacle when dealing with long documents or high-resolution images. In this work, we study the self-attention mechanism by analyzing the distribution of the attention matrix and its concentration ability. Furthermore, we propose instruments to measure these quantities and introduce a novel self-attention mechanism, Linear Log-Normal Attention, designed to emulate the distribution and concentration behavior of the original self-attention. Our experimental results on popular natural language benchmarks reveal that our proposed Linear Log-Normal Attention outperforms other linearized attention alternatives, offering a promising avenue for enhancing the scalability of transformer models.

Linear Log-Normal Attention with Unbiased Concentration

TL;DR

Addressing the quadratic cost of self-attention, the paper analyzes the distribution and concentration of the SA matrix and introduces Linear Log-Normal Attention (LLN), a linear-time mechanism built from exponential feature maps to mimic SA's log-normal distribution. LLN employs moment matching to align its variance with SA and uses a temperature parameter to control concentration, supported by entropy and spectral-gap analyses that distinguish unbiased from biased concentration. Empirically, LLN matches SA performance on NLP benchmarks while offering substantial scalability, and a block-diagonal augmentation further strengthens short-range interactions. Overall, the work provides a rigorous framework for scalable attention by coupling distributional mimicry with concentration metrics, and it releases code to enable broader use.

Abstract

Transformer models have achieved remarkable results in a wide range of applications. However, their scalability is hampered by the quadratic time and memory complexity of the self-attention mechanism concerning the sequence length. This limitation poses a substantial obstacle when dealing with long documents or high-resolution images. In this work, we study the self-attention mechanism by analyzing the distribution of the attention matrix and its concentration ability. Furthermore, we propose instruments to measure these quantities and introduce a novel self-attention mechanism, Linear Log-Normal Attention, designed to emulate the distribution and concentration behavior of the original self-attention. Our experimental results on popular natural language benchmarks reveal that our proposed Linear Log-Normal Attention outperforms other linearized attention alternatives, offering a promising avenue for enhancing the scalability of transformer models.
Paper Structure (30 sections, 9 theorems, 49 equations, 10 figures, 5 tables)

This paper contains 30 sections, 9 theorems, 49 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Background on Self-Attention
  4. Linearized Attention
  5. Dissecting Softmax Attention
  6. Characterizing distribution of Softmax Attention
  7. Analyzing Self-Attention through Markov Chain Perspective
  8. Entropy and Attention Concentration
  9. Spectral Gap and Attention Concentration
  10. Design of Linearized Attention
  11. Linear Log-Normal Attention
  12. The overall architecture
  13. Experiments
  14. Speed and memory consumption
  15. Conclusion
  16. ...and 15 more sections

Key Result

Proposition 3.1

Let ${\pmb{q}}$ and ${\pmb{k}}$ be Gaussian vectors, where $q_i\sim \mathcal{N}(0,\sigma_q^2)$ and $k_j\sim \mathcal{N}(0,\sigma_k^2)$, $\forall i,j$. Then, for moderate values of $\sigma_q^2, \sigma_k^2$ and large enough $N$ the distribution of ${\pmb{P}}^{\text{(SM)}}$ can be approximated by a log

Figures (10)

  • Figure 1: Temperature (left), entropy (center), and spectral gap (right) during training of the small RoBERTa model with a single head per layer in every training step (X-axis).
  • Figure 2: Comparison of entropy (left) and spectral gap (right) for various types of attention kernels. The figure shows that the entropy and spectral gap of the LLN Attention with the moment matching is similar to those of the SA.
  • Figure 3: LLN Transformer layer architecture.
  • Figure 4: A block diagram of the computational complexity for the Softmax Attention and Linearized Attention.
  • Figure 5: (a) The variance and mean of the SA matrix with respect to the input variance. Measurements perfectly match theoretical estimation. (b) The variance of the SA and LLN Attention before and after performing the moment matching procedure.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Proposition 4.1
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Theorem A.3
  • ...and 7 more