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Published equation contexts

E[tokens per round]=1−αγ+11−α\mathbb{E}[\text{tokens per round}] = \frac{1-\alpha^{\gamma+1}}{1-\alpha}

Why this formula appears here

The size of the win has a clean shape. Model the draft’s acceptance probability as α\alpha per token, roughly constant and independent across the γ\gamma tokens proposed in a round — an idealization real traffic does not fully satisfy, but a useful one for seeing the ceiling. The expected number of tokens accepted per verification round is then E[tokens per round]=1−αγ+11−α\mathbb{E}[\text{tokens per round}] = \frac{1-\alpha^{\gamma+1}}{1-\alpha}. This rises with both α\alpha and γ\gamma , but with steeply diminishing returns in γ\gamma for any α\alpha below one: drafting fifty tokens ahead does not buy anywhere near fifty accepted tokens, because the marginal proposal deep into a long draft is unlikely to be exactly what the target would have generated. The bottleneck decode…

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αγ+1\alpha^{\gamma+1}

Symbol α^gamma+1

α^gamma+1 occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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E[tokens per round]=1−αγ+11−α.\mathbb{E}[\text{tokens per round}] = \frac{1-\alpha^{\gamma+1}}{1-\alpha}.

Equation 18 · Inference Economics

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The size of the win has a clean shape. Model the draft’s acceptance probability as α\alpha per token, roughly constant and independent across the γ\gamma tokens proposed in a round — an idealization real traffic does not fully satisfy, but a useful one for seeing the ceiling. The expected number of tokens accepted per verification round is then E[tokens per round]=1−αγ+11−α\mathbb{E}[\text{tokens per round}] = \frac{1-\alpha^{\gamma+1}}{1-\alpha}. This rises with both α\alpha and γ\gamma , but with steeply diminishing returns in γ\gamma for any α\alpha below one: drafting fifty tokens ahead does not buy anywhere near fifty accepted tokens, because the marginal proposal deep into a long draft is unlikely to be exactly what the target would have generated. The bottleneck decode…

Meanings in this article

  • E\mathbb{E}: The expected value operator: the probability-weighted average of the quantity inside its brackets.
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