Budget-aware Test-time Scaling via Discriminative Verification

TL;DR

This work tackles the challenge of efficient test-time scaling for complex reasoning with LLMs by advocating budget-aware discriminative verification and integrating it with self-consistency. The authors introduce hybrid methods, namely Weighted Self-Consistency (WSC) and Pessimistic Verification (PV), which combine consensus signals with a lightweight discriminative verifier trained via a Bradley–Terry ranking loss. Their results show that these hybrids often outperform generative verification under realistic compute budgets, achieving notable gains on tasks like AIME2025 while requiring far less compute and latency. The study also analyzes FLOPs and latency to demonstrate practical advantages, scales the verifier and solver sizes, and demonstrates robustness across multiple benchmarks, making hybrid discriminative verification a compelling alternative for real-world deployment.

Abstract

Test-time scaling is a powerful strategy for boosting the performance of large language models on complex reasoning tasks. While state-of-the-art approaches often employ generative verifiers to select the best solution from a pool of candidates, this method incurs prohibitive computational costs, limiting its practicality. In this work, we shift the focus to a more budget-aware paradigm: discriminative verification. We conduct a thorough empirical analysis and demonstrate that while discriminative verifiers may underperform in isolation, combining them with self-consistency in a hybrid approach creates a powerful and efficient test-time scaling mechanism. Notably, under a fixed compute budget, this hybrid approach surpasses state-of-the-art generative verification by a significant margin: achieving up to 15.3\% higher accuracy on AIME2025. Our findings establish that for practical, real-world applications, budget-aware scaling with discriminative verifiers is not only a "free" upgrade over self-consistency, but also a more effective and efficient alternative to costly generative techniques. Code is available at https://github.com/wang-research-lab/verification.

Figures (7)

• Figure 1: Hybrid discriminative verification techniques (e.g., weighted self-consistency (WSC) welleck2024decoding and pessimistic verification (PV) shi2025heimdalltesttimescalinggenerative) outperform generative pessimistic verification (GPV) under equalized compute budgets of less than 22.5 minutes (shaded region). For example, at latency budgets of 13.8 minutes and 15.7 minutes, hybrid discriminative verification can outperform generative verification by 15.3% and 2.8%, respectively. $N$ is doubled at each point along the x-axis. For GPV, each solution is verified twice ($M=2$).
• Figure 2: Blue: The loss decreases over one epoch of training. Red: The score margin, i.e., the difference in score assigned to correct solutions and incorrect solutions on average across a global batch, increases during training. Together, these indicate that the discriminative verifier learns to discriminate between correct and incorrect solutions.
• Figure 3: Accuracy vs. FLOPs on AIME2025 under equalized compute budgets. Each subplot varies the number of verifications per candidate solution $(M)$. Along each curve, successive points correspond to doubling the number of candidate solutions $(N)$. The shaded region highlights the FLOPs budgets where hybrid discriminative verification techniques strictly outperform generative verification under equalized compute budgets.
• Figure 4: Accuracy rates on AIME 2024/2025 for various discriminative verification methods across four solver sizes for several values of $N$. Pass@$N$ and SC@$N$ are included as baselines.
• Figure 5: Left: Unlike BoN, hybrid techniques show consistent but diminishing improvements on AIME2024 from increasing the number of candidate results $N$ sampled from DeepSeek-R1-Distill-Qwen-32B. Right: The performance of DeepSeek-R1-Distill-Qwen-32B on AIME2024 scales logarithmically with the reasoning budget regardless of verification method. Here, $N=32$.