Matt sat on the chair LM p(· | context) couch 2.24 0.131 chair 1.73 + 2.0 0.581 floor 1.01 + 2.0 0.283 cat −1.98 0.002 table −3.66 + 2.0 0.003 potato −6.23 0.000 logits + δ probabilities Pseudo-Random Function PRF(key, last $k$ tokens) → G ∪ R

PRF hash of previous $k$ tokens splits the vocabulary into green ($\gamma|V|$) and red. Add $\delta$ (here $\delta=2$) to green logits before softmax. Most likely token shifts: couch → chair.

Soft red/green watermark:

$\tilde p_v^{(t)} = \dfrac{\exp(\ell_v^{(t)} + \delta\,\mathbb{1}[v\in G])}{\sum_{j} \exp(\ell_j^{(t)} + \delta\,\mathbb{1}[j\in G])}$

Hyperparameters:

  • $\gamma$: green-list fraction
  • $\delta$: green-logit bias
  • $k$: hash window over previous $k$

Kirchenbauer et al., ICML 2023, § 3