A Simple and Provable Scaling Law for the Test-Time Compute of Large
  Language Models
Authors: name: Yanxi Chen, name: Xuchen Pan, name: Yaliang Li, name: Bolin Ding, name: Jingren Zhou

We propose a general two-stage algorithm that enjoys a provable scaling law
for the test-time compute of large language models (LLMs). Given an input
problem, the proposed algorithm first generates N candidate solutions, and
then chooses the best one via a multiple-round knockout tournament where each
pair of candidates are compared for K times and only the winners move on to
the next round. In a minimalistic implementation, both stages can be executed
with a black-box LLM alone and nothing else (e.g., no external verifier or
reward model), and a total of N times (K + 1) highly parallelizable LLM
calls are needed for solving an input problem. Assuming that a generated
candidate solution is correct with probability p_{gen} > 0 and a
comparison between a pair of correct and incorrect solutions identifies the
right winner with probability p_{comp} > 0.5 (i.e., better than a
random guess), we prove theoretically that the failure probability of the
proposed algorithm decays to zero exponentially with respect to N and K:
$P(final output is incorrect) le (1 - p_{gen})^N +
lceil log_2 N rceil e^{-2 K (p_{comp} - 0.5)^2}.$ Our empirical
results with the challenging MMLU-Pro benchmark validate the technical
assumptions, as well as the efficacy of the proposed algorithm and the gains
from scaling up its test-time compute.