tl;dr: The bias term for preventing attention sink in GPT-OSS may have other effects than just preventing attention sink. (made public on 2026-01-22)

Notation

\[\begin{alignat}{4} &\bm{X} &:= & \begin{bmatrix} \bm{x}_1\\ \vdots\\ \bm{x}_n \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{n \times d}\\ &\bm{W}^O &:= & \begin{bmatrix} \bm{W}^O_1\\ \vdots\\ \bm{W}^O_H \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{d \times d}\\ &\bm{W}^Q &:= & \begin{bmatrix} \bm{W}^Q_1 & \cdots & \bm{W}^Q_H \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{d \times d}& \label{eq:wq_split}\\ &\bm{W}^K &:= & \begin{bmatrix} \bm{W}^K_1 & \cdots & \bm{W}^K_H \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{d \times d}& \label{eq:wk_split}\\ &\bm{W}^V &:= & \begin{bmatrix} \bm{W}^V_1 & \cdots & \bm{W}^V_H \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{d \times d}&\label{eq:wv_split}\\ &\bm{b}^Q &:= & \begin{bmatrix} \bm{b}^Q_1 & \cdots & \bm{b}^Q_H \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{d}& \label{eq:bq_split}\\ &\bm{b}^K &:= & \begin{bmatrix} \bm{b}^K_1 & \cdots & \bm{b}^K_H \end{bmatrix} &\hspace{1em}\in& \mathbb{R}^{d}& \label{eq:bk_split}\\ &\bm{b}^V &:= & \begin{bmatrix} \bm{b}^V_1 & \cdots & \bm{b}^V_H \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{d}& \\ &\bm{I} &:= & \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{d\times d}& \\ &\bm{1} &:= & \begin{bmatrix} 1 & \cdots & 1 \end{bmatrix} &\hspace{1em}\in &\mathbb{R}^{d} \end{alignat}\]

Self-Attention

Let query, key, value transformations of each head $h$ be expressed as follows:

\[\begin{align} \bm{q}_h(\bm{x}) &:= \bm{x}\bm{W}_h^Q + \bm{b}_h^Q\\ \bm{k}_h(\bm{x}) &:= \bm{x}\bm{W}_h^K + \bm{b}_h^K\\ \bm{v}_h(\bm{x}) &:= \bm{x}\bm{W}_h^V + \bm{b}_h^V\\ \end{align}\]

Let attention weight from token position $i$ to $j$ ($i \ge j$) in head $h$ be expressed as follows:

\[\begin{align} \alpha_{i, j, h} &= \frac{\exp(s_{i, j, h})}{\exp(b^{S}_h)+\sum_{j'} \exp(s_{i, j', h})}\\ s_{i, j, h} &:= \frac{\bm{q}_h(\bm{x}_i)\bm{k}_h(\bm{x}_j)^\top}{\sqrt{d'}} \end{align}\]

where $d’ = d/H$ is the dimension of each head, and $b^{S}_h$ is a learned scalar parameter introduced in GPT-OSS for preventing attention sink.

Now, let the attention weight assigned to the sink be expressed as follows:

\[\alpha_{i, \text{sink}, h} := \frac{\exp(b^{S}_h)}{\exp(b^{S}_h)+\sum_{j'} \exp(s_{i, j', h})}\]

Due to the presence of $b^{S}_h$, the following holds:

\[\begin{align} &1 = \alpha_{i, \text{sink}, h} + \sum_j \alpha_{i, j, h}\\ &\Leftrightarrow \sum_j \alpha_{i, j, h} = 1 - \alpha_{i, \text{sink}, h} \end{align}\]

The output of Attention layer of an causal model at position $i$ can be expressed as follows:

\[\begin{align} \text{ATTN}(i, \bm{X}) &:=\left[\text{head}_1(i, \bm{X})\hspace{0.5em}\cdots\hspace{0.5em}\text{head}_H(i, \bm{X})\right] \bm{W}^O + \bm{b}^O\\ &=\sum_{h=1}^H \text{head}_h(i, \bm{X})\bm{W}^O_h + \bm{b}^O\\ &=\sum_{h=1}^H \left(\sum_{j=1}^i \alpha_{i, j, h} \bm{v}_h(\bm{x}_j)\right)\bm{W}^O_h + \bm{b}^O\\ &=\sum_{h=1}^H \left(\sum_{j=1}^i \alpha_{i, j, h} \left(\bm{x}_j\bm{W}^V_h + \bm{b}^V_h\right)\right)\bm{W}^O_h + \bm{b}^O\\ &=\sum_{h=1}^H \left(\sum_{j=1}^i \alpha_{i, j, h} \left(\bm{x}_j\bm{W}^V_h + \bm{b}^V_h\right)\bm{W}^O_h\right) + \bm{b}^O\\ &=\sum_{h=1}^H \left(\sum_{j=1}^i \alpha_{i, j, h} \left(\bm{x}_j\bm{W}^V_h\bm{W}^O_h + \bm{b}^V_h\bm{W}^O_h\right)\right) + \bm{b}^O\\ &= \sum_{h=1}^H \sum_{j=1}^i \alpha_{i, j, h}\bm{x}_j\bm{W}^V_h\bm{W}^O_h + \sum_{h=1}^H \left(\sum_{j=1}^i \alpha_{i, j, h}\bm{b}^V_h\bm{W}^O_h\right) + \bm{b}^O\\ &= \sum_{h=1}^H \sum_{j=1}^i \alpha_{i, j, h}\bm{x}_j\bm{W}^V_h\bm{W}^O_h + \sum_{h=1}^H \left(\sum_{j=1}^i \alpha_{i, j, h}\right)\bm{b}^V_h\bm{W}^O_h + \bm{b}^O\\ &= \sum_{h=1}^H \sum_{j=1}^i \alpha_{i, j, h}\bm{x}_j\bm{W}^V_h\bm{W}^O_h + \sum_{h=1}^H \left(1-\alpha_{i, \text{sink}, h}\right)\bm{b}^V_h\bm{W}^O_h + \bm{b}^O\\ &= \sum_{h=1}^H \sum_{j=1}^i \alpha_{i, j, h}\bm{x}_j\bm{W}^V_h\bm{W}^O_h + \sum_{h=1}^H\bm{b}^V_h\bm{W}^O_h - \sum_{h=1}^H\alpha_{i, \text{sink}, h}\bm{b}^V_h\bm{W}^O_h + \bm{b}^O\\ &= \sum_{h=1}^H \sum_{j=1}^i \alpha_{i, j, h}\bm{x}_j\bm{W}^V_h\bm{W}^O_h + \sum_{h=1}^H\alpha_{i, \text{sink}, h}(-\bm{b}^V_h\bm{W}^O_h)+ \sum_{h=1}^H\bm{b}^V_h\bm{W}^O_h + \bm{b}^O\\ &= \sum_{h=1}^H \sum_{j=1}^i \alpha_{i, j, h}\bm{x}_j\bm{W}^V_h\bm{W}^O_h+ \sum_{h=1}^H\alpha_{i, \text{sink}, h}(-\bm{b}^V_h\bm{W}^O_h) + \bm{b}^V\bm{W}^O + \bm{b}^O\\ \end{align}\]

Thus, in GPT-OSS, the newly introduced attention sink bias $b^{S}_h$

  • makes the attention weights sum to less than 1 for non-sink tokens, and
  • changes the intensity of the bias term of the value transformation based on the attention weight assigned to the sink.