QuasiMoTTo: Quasi-Monte Carlo Test-Time Scaling

2026-07-01

AI Summary

Traditional parallel sampling in language model inference suffers from substantial computational redundancy due to the independence of generated samples. This work proposes the first framework integrating quasi-Monte Carlo (QMC) methods into autoregressive language model parallel sampling, leveraging inverse CDF reparameterization to produce marginally exact yet mutually correlated samples. This approach preserves distributional correctness while significantly enhancing sampling efficiency. To rigorously evaluate such correlated samplers, the authors further introduce an unbiased bootstrap estimator. Empirical results demonstrate that the proposed method achieves equivalent pass@k accuracy using 25%–47% fewer samples across four reasoning benchmarks and matches the performance of independent sampling in GRPO reinforcement learning with only half the training steps.

Abstract

Scaling inference compute, by generating many parallel attempts per problem, is a costly but reliable lever for improving language model capabilities. By default these attempts are generated independently, wasting inference compute on redundant solutions. This waste seems unavoidable. After all, independence is what makes parallel sampling trivial to scale. However, this tradeoff is not fundamental: there is a rich design space of samplers that generate correlated but exact samples entirely in parallel. We explore this design space as an avenue for improving sample efficiency in scaling inference compute and reinforcement learning (RL). Concretely, we introduce QuasiMoTTo, which uses correlated samples as a drop-in replacement for i.i.d. samples. To generate these samples, QuasiMoTTo uses a reparameterization of autoregressive sampling as inverse-CDF sampling and draws the underlying uniforms with quasi-Monte Carlo (QMC); because QMC spreads the uniforms out more evenly than i.i.d., the resulting samples cover the output space with far less redundancy. Even though the batch is correlated, each sample is marginally distributed according to the language model, so we can use the batch for policy-gradient training. Our empirical analysis focuses on understanding how efficiently QuasiMoTTo can turn compute into performance. To evaluate correlated samplers, whose dependence breaks standard pass@k estimators, we first develop an unbiased bootstrap estimator. Across four reasoning benchmarks, QuasiMoTTo matches i.i.d. pass@k accuracy with 25-47% fewer samples. Strikingly, QuasiMoTTo often saturates an upper bound on pass@k that holds for any marginal-preserving sampler. We also apply QuasiMoTTo to policy-gradient RL (GRPO) where it matches i.i.d. performance with 50% fewer training steps. These gains come from higher coverage, which yields a stronger learning signal per batch.

Problem

• sample efficiency
• inference scaling
• redundant sampling
• language models
• parallel sampling

Innovation

• Quasi-Monte Carlo
• test-time scaling
• correlated sampling
• inverse-CDF sampling
• sample efficiency