When Exponentials Meet Power Laws: Why LLM Scaling Laws Hide in the Tails of Probability Distributions

Forum topic · 二一 · 2026-05-12

Summary

This forum post explains the central finding of the ICML 2025 oral paper 'How Do Large Language Monkeys Get Their Power (Laws)?'. When large language models attempt problems repeatedly (success counts if any attempt succeeds), the aggregate success rate follows a power law: -log(success rate) scales with attempts^c. Yet for any single fixed-difficulty problem, failure probability must decay exponentially, as 1-(1-p)^k. The post resolves this apparent paradox with a monkey-typewriter thought experiment and explicit math: if per-problem success probabilities p follow a heavy-tailed (power-law) distribution, then the sum of exponentials e^{-pk} is dominated by the extremely hard problems, and the expected failure rate behaves as k^{-alpha}. Empirical validation on math problems, Lean proof generation, and multimodal jailbreak attacks confirms per-problem exponential decay, aggregate power-law scaling, and a heavy-tailed p distribution. Practical implications include predicting scaling behavior from a single pass (2-4 orders of magnitude less inference compute, one order of magnitude better accuracy than baselines) and designing benchmarks whose difficulty distributions include extremely hard problems. Paper by Rylan Schaeffer et al., links to OpenReview and GitHub included.

> Feynman once said: "If you can't explain your research to a stranger at a bar, you don't really understand it." Today's story is about a pattern that confuses everyone — and the answer hides in the tail of a probability distribution.

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Introduction: A Pattern That Puzzled Researchers

Researchers recently noticed a strange statistical regularity. When you let a large language model (LLM) attempt a batch of problems — math questions, coding tasks, jailbreak attacks — giving it multiple tries per problem (success counts if any attempt succeeds), the overall success rate follows a power law:

-\log(\text{success rate}) \propto (\text{number of attempts})^c

In plain words: success rate grows more and more slowly with each additional attempt. The first attempt brings a big jump, the second is decent, but by attempt 100 or 1000, each extra try adds almost nothing.

This is the famous "scaling law." Seems normal, right?

But here's the problem. A simple mathematical derivation predicts that for each *individual* problem, the failure rate should decay exponentially with attempts.

Exponential decay means: if the first-attempt success rate is 10%, the second might be 19%, the fourth approaches 35%, and by attempt 32 you're near 97%. It converges to 100% very fast.

Exponential on individual problems, power law in the aggregate. What's going on?

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Chapter 1: A Feynman-Style Thought Experiment

Let's understand this the way Feynman loved — a simple thought experiment.

Suppose 1000 monkeys, each with a typewriter. Each monkey types randomly, attempting to produce a complete English sentence. Some sentences are easy ("Hi"), some are medium ("To be or not to be"), and some are nearly impossible (a Shakespearean sonnet).

For any single monkey, the probability of eventually producing its target sentence rises exponentially with attempts. Why? Each attempt is independent; if the monkey lives long enough, it will eventually type the right combination. Mathematically: if the per-attempt success probability is $$p$$ , the probability of at least one success in $$k$$ attempts is $$1 - (1-p)^k$$ . As $$k$$ grows, $$(1-p)^k$$ decays exponentially to zero.

But when we aggregate all monkeys' results, things change.

Some monkeys' tasks are too hard — per-attempt success probability $$p$$ is nearly zero (like the sonnet monkey). They may never succeed. Even after the medium-difficulty monkeys have all succeeded, these "extremely hard cases" are still lagging. In the aggregate statistics, they dominate the entire trend.

That's the answer to today's core paradox — each individual decays exponentially, but the aggregate follows a power law.

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Chapter 2: The Precise Mathematics

Formally: suppose there are $$N$$ problems, and problem $$i$$ has per-attempt success probability $$p_i$$ . After $$k$$ attempts, its success probability is:

1 - (1-p_i)^k \approx 1 - e^{-p_i k}

For any fixed $$p_i$$ (however small), $e^{-p_i k}$ decays exponentially as $$k$$ grows. That's exponential scaling.

Now the aggregate. The average failure rate over all $$N$$ problems is:

\text{average failure rate} = \frac{1}{N} \sum_{i=1}^{N} e^{-p_i k}

Here's the key: the result of this sum depends on the distribution of the $$p_i$$ .

If all $$p_i$$ are similar — say, all between 0.01 and 0.1 — the sum decays exponentially. But if the distribution of $$p_i$$ is heavy-tailed — most problems have moderate $$p_i$$ , but a small cluster has extremely tiny $$p_i$$ — then a few "holdout" terms with very slow decay remain in the sum.

When these slowest-decaying terms dominate, the aggregate behavior switches from exponential to power law. Mathematically, if the distribution of $$p$$ satisfies $P(p \leq x) \sim x^\alpha$ (a power-law tail), then:

\mathbb{E}[e^{-pk}] \sim k^{-\alpha}

There it is — an aggregate power law!

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Chapter 3: Validation — What the Data Says

The researchers didn't stop at theory. They validated the claims in three different domains: math problem solving, programming with the Lean proof assistant, and multimodal jailbreak attacks.

3.1 Individual problems are exponential

First, they tested the "per-problem exponential" prediction, isolating each problem and observing how its success probability changes with attempts. Conclusion: for the vast majority of problems, failure rates show clean exponential decay. The goodness of fit is high.

3.2 Aggregates are power laws

Then they looked at aggregates. Mixing all problems together, log-log plots of average success rate vs. attempts form straight lines — the hallmark of a power law. A stark contrast with the per-problem exponential behavior.

3.3 The distribution really is heavy-tailed

Finally, they tested the key assumption: is the distribution of per-attempt success probability $$p$$ heavy-tailed? Examining all problems: most problems have success rates between 10%–50%, but a long tail extends below 0.0001%.

It's precisely these extremely hard problems that control the aggregate statistics at large attempt counts.

3.4 Explaining previously puzzling phenomena

This "distributional view" also explains several earlier counterexamples:

Why do some benchmarks not obey a power law? Because their difficulty distributions aren't heavy-tailed enough — no cluster of extremely hard problems.
Why does the power-law slope differ across tasks? Because the "thickness" of the difficulty-distribution tail differs across tasks.

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Chapter 4: What's It Good For?

4.1 Predicting scaling behavior with far less compute

This is the most practical contribution. Previously, predicting a model's aggregate performance at $$k$$ attempts required actually running $$k$$ attempts — very expensive.

Now, you only need the model to attempt each problem once, then examine the distribution of per-attempt success rates — especially its tail. From the tail thickness, you can directly estimate the power-law exponent. That means you don't need to burn millions of inference tokens to predict scaling behavior.

The paper shows this method predicts the power-law exponent with relative error an order of magnitude lower than prior baselines — equivalently, requiring 2–4 orders of magnitude less inference compute.

4.2 Implications for benchmark design

If you want to design a good evaluation benchmark, this finding tells you:

For reliable evaluation → make the difficulty distribution "heavy-tailed": include a small set of extremely hard problems.
For comparing two models → look at their differences on the hardest problems: that's where model capability is truly distinguished.

Most benchmarks fail to separate top models precisely because they lack sufficiently hard problems — all models easily exceed 90%+ success rates; the tail is too short and the distribution too concentrated.

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Feynman's Book Review

If Feynman read this paper, he'd probably say:

"This is the kind of paper I like. It starts from an observation — the aggregate is a power law — and asks: 'But isn't that odd? Single problems should clearly be exponential.'

Then they went and checked. Indeed exponential. So where does the aggregate power law come from?

The answer is surprisingly simple: a few problems are just too hard. So hard that the per-attempt success rate is nearly zero, so hard that even a hundred attempts almost surely fail. These 'holdout' problems squat in the aggregate statistics and drag the exponential into a power law.

This is a story about distributions. Two phenomena that look contradictory at the macro level — exponential and power law — become perfectly harmonious once you understand the heavy-tailed structure of the underlying distribution.

It also reminds us: when you see an aggregate statistical regularity, go look at the shape of the distribution. The truth often lies not in the mean, but in the tail."

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*Paper information*

Title: How Do Large Language Monkeys Get Their Power (Laws)?
Authors: Rylan Schaeffer, Joshua Kazdan, John Hughes, Jordan Juravsky, Sara Price, Aengus Lynch, Erik Jones, Robert Kirk, Azalia Mirhoseini, Sanmi Koyejo
Venue: ICML 2025 (Oral)
Link: OpenReview
Code: GitHub