When To Solve, When To Verify: Compute-Optimal Problem Solving and Generative Verification for LLM Reasoning

TL;DR

This work compares Generative Reward Models (GenRM) and Self-Consistency (SC) under a fixed inference budget to determine compute-optimal strategies for LLM reasoning. It shows SC is more compute-efficient at practical budgets, while GenRM can surpass SC only at substantially larger budgets, requiring much more compute. The authors introduce a compute-matched framework and derive inference scaling laws, revealing that the optimal number of solutions grows faster with budget than the number of verifications (S_opt ∝ C^{0.57}, V_opt ∝ C^{0.39}), providing actionable budgeting guidance. The findings are validated across multiple model families and math-reasoning tasks, offering concrete recommendations for deploying scalable reasoning with verification in real-world settings.

Abstract

Scaling test-time compute has emerged as a key strategy for enhancing the reasoning capabilities of large language models (LLMs), particularly in tasks like mathematical problem-solving. A traditional approach, Self-Consistency (SC), generates multiple solutions to a problem and selects the most common answer via majority voting. Another common method involves scoring each solution with a reward model (verifier) and choosing the best one. Recent advancements in Generative Reward Models (GenRM) reframe verification as a next-token prediction task, enabling inference-time scaling along a new axis. Specifically, GenRM generates multiple verification chains-of-thought to score each solution. Under a limited inference budget, this introduces a fundamental trade-off: should you spend the budget on scaling solutions via SC or generate fewer solutions and allocate compute to verification via GenRM? To address this, we evaluate GenRM against SC under a fixed inference budget. Interestingly, we find that SC is more compute-efficient than GenRM for most practical inference budgets across diverse models and datasets. For instance, GenRM first matches SC after consuming up to 8x the inference compute and requires significantly more compute to outperform it. Furthermore, we derive inference scaling laws for the GenRM paradigm, revealing that compute-optimal inference favors scaling solution generation more aggressively than scaling the number of verifications. Our work provides practical guidance on optimizing test-time scaling by balancing solution generation and verification. The code is available at https://github.com/nishadsinghi/sc-genrm-scaling.

Figures (10)

• Figure 1: Left: The prominent approach is to compare GenRM and Self-Consistency (SC) at a fixed number of solutions, suggesting that GenRM is more efficient as it matches SC with fewer solutions. Right: When evaluated under a fixed compute budget, including verification costs, SC outperforms GenRM at lower budgets, using up to $8\times$ less compute, while GenRM excels at higher budgets. Each curve corresponds to a fixed number of verifications; the number of solutions is doubled at each point along the x-axis. The solutions are generated by Llama-3.1-8B-Instruct grattafiori2024llama, which also performs verifications after being fine-tuned as GenRM, on the MATH dataset hendrycks2021measuring.
• Figure 2: Compute-Matched Analysis. Given a fixed inference budget $B$, our analysis (b) compares the performance of (a) scaling the number of solutions ($S = B$) with Self-Consistency vs. (c) generating fewer solutions ($S = B/V$) while relying on verifications ($V$) using Generative Reward Models.
• Figure 3: Left: Relative improvement achieved by Llama-3.1-8B-Instruct GenRM-FT (32 verifications) over SC for different difficulty levels in MATH. Hard problems benefit more from GenRM-FT, with up to 30% relative improvement over SC. Right: Comparing GenRM-FT against GenRM-Base, we find that GenRM-FT consistently performs better, requiring much less compute to match the performance of GenRM-Base. This highlights the importance of high-quality verifications.
• Figure 4: (Left) Evaluation of GenRM-FT (Llama-3.1-8B trained on MATH) generalizing to AIME24. GenRM-FT provides significant improvements over Self-Consistency (SC) on these harder problems, demonstrating its generalization ability, though it requires substantially more compute to outperform SC. (Right) Comparison of GenRM-Base versus SC for an RL-tuned QwQ-32B model. This confirms previous observations: SC performs better at lower budgets, while GenRM shines at higher budgets.We extrapolate the SC curves (dashed lines) because their performance saturates beyond a certain point.
• Figure 5: Comparing GenRM-Base with Llama-3.3-70B-Instruct on (a) MATH, and (b) GPQA-Diamond, respectively. These results highlight that across model sizes and reasoning domains, GenRM outperforms Self-consistency (SC) only at high compute budgets. We extrapolate the SC curve (horizontal dashed line) as its performance saturates after a point.