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How Do Large Language Monkeys Get Their Power (Laws)?
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Shows that aggregate power-law scaling of repeated-sampling success rates arises from a power-law left tail in the distribution of single-attempt success probabilities, even though each problem scales exponentially, and uses this to build a distributional estimator for forecasting the exponent.
Problem
Repeated-sampling studies (Large Language Monkeys, Best-of-N jailbreaking) report that negative log average success rate scales as a power law in the number of attempts k, yet per-problem math implies exponential decay; the origin of this aggregate power law is unexplained.
Approach
Aggregate power-law scaling emerges when the distribution over problems of single-attempt success probabilities has a power-law-like left tail near zero (density ~ p^{b-1} gives -log pass@k ~ k^{-b}); this distributional view explains deviations from power laws and yields a more sample-efficient exponent estimator.
Key results
- Per-problem -log(pass_i@k) decays exponentially in k on MATH (Pythia) and HarmBench (frontier models), while the aggregate decays as a power law.
- The distributional estimator achieves roughly an order of magnitude lower relative error in exponent estimation than least squares, equivalent to ~2-4 orders of magnitude less inference compute.
- Llama 3 8B IT lacks a heavy left tail in single-attempt ASR, explaining why it does not show aggregate power-law scaling.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the distributional estimator instead of least-squares regression to estimate the power-law exponent | Power-law exponent estimation · Synthetic backtesting | relative error: approximately an order of magnitude lower relative error | Helped |
| Use the distributional estimator instead of least-squares regression | Inference-compute requirement for exponent estimation · Synthetic backtesting | inference compute: ${\sim}2-4$ orders of magnitude less inference-compute | Helped |
| Fit the distribution of single-attempt success rates and simulate aggregate performance | Power-law exponent estimation · Large Language Monkeys and Best-of-N Jailbreaking | exponent agreement with least-squares estimator | No clear effect |
| Have a power-law-like left tail in the distribution of single-attempt success probabilities | Repeated-sampling success rates across benchmark problems · Aggregate repeated-attempt scaling | negative log average success rate | Helped |
| Use a single-attempt success-probability distribution without a heavy left tail | Llama 3 8B IT jailbreak success · HarmBench | power-law scaling | Hurt |
Surprising findings
- Llama 3 8B IT could be jailbroken on every prompt within budget, so its aggregate curve does not follow a power law.
- Estimating pass@k as 1-(1-pass@1)^k is biased, so naive plug-in estimates are misleading.
Limitations
- Best-of-N jailbreaking samples were not strictly independent (minibatches, early stopping), a deviation the authors acknowledge in an appendix.
- Theoretical results rely on asymptotic large-k regimes and mild regularity assumptions.
- Empirical validation uses only two benchmark/setting families (MATH with Pythia, HarmBench jailbreaks).
Experimental setup
Datasets: MATH HarmBench
Benchmarks: MATH HarmBench
Metrics: negative log pass@k negative log ASR@k relative error of power law exponent
Baselines: least squares power law estimator
Builds on: Large Language Monkeys Best-of-N Jailbreaking unbiased pass@k estimator
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