Test-time Verification via Optimal Transport: Coverage, ROC, & Sub-optimality

TL;DR

This paper models test-time verification in large language models as an optimal transport problem, revealing three regimes of sub-optimality and coverage interaction, supported by empirical results across multiple models.

Contribution

It introduces a unified transport framework to analyze the interplay of coverage, verifier ROC, and sub-optimality in test-time scaling, and characterizes three distinct regimes.

Findings

Identification of three regimes: transport, policy improvement, saturation.

Analysis of sequential and batched sampling algorithms and their complexities.

Empirical validation with Qwen, Llama, and Gemma models.

Abstract

While test-time scaling with verification has shown promise in improving the performance of large language models (LLMs), the role of the verifier and its imperfections remain underexplored. The effect of verification manifests through interactions of three quantities: (i) the generator's coverage, (ii) the verifier's region of convergence (ROC), and (iii) the sampling algorithm's sub-optimality. Though recent studies capture subsets of these factors, a unified framework quantifying the geometry of their interplay is missing. We frame verifiable test-time scaling as a transport problem. This characterizes the interaction of coverage, ROC, and sub-optimality, and uncovers that the sub-optimality--coverage curve exhibits three regimes. A transport regime -- where sub-optimality increases with coverage, a policy improvement regime -- where sub-optimality may decrease with coverage, depending on the verifier's ROC, and a saturation regime -- where sub-optimality plateaus, unaffected by coverage. We further propose and analyze two classes of sampling algorithms -- sequential and batched, and examine how their computational complexities shape these trade-offs. Empirical results with Qwen, Llama, and Gemma models corroborate our theoretical findings.