2510.20272
Model: nemotron-free
# Limits of PRM-Guided Tree Search for Mathematical Reasoning with LLMs
**Authors**:
- Tristan Cinquin (University of TĆ¼bingen)
- &Geoff Pleiss (University of British Columbia & Vector Institute)
- &Agustinus Kristiadi (Western University & Vector Institute)
> Work done while interning at the Vector Institute.
\workshoptitle
MATH-AI
Abstract
While chain-of-thought prompting with Best-of-N (BoN) selection has become popular for mathematical reasoning in large language models (LLMs), its linear structure fails to capture the branching and exploratory nature of complex problem-solving. In this work, we propose an adaptive algorithm to maximize process reward model (PRM) scores over the intractable action space, and investigate whether PRM-guided tree search can improve mathematical reasoning by exploring multiple partial solution paths. Across $23$ diverse mathematical problems using Qwen2.5-Math-7B-Instruct with its associated PRM as a case study, we find that: (1) PRM-guided tree search shows no statistically significant improvements over BoN despite higher costs, (2) Monte Carlo tree search and beam search outperform other PRM-guided tree search methods, (3) PRMs poorly approximate state values and their reliability degrades with reasoning depth, and (4) PRMs generalize poorly out of distribution. This underperformance stems from tree searchās greater reliance on unreliable PRM scores, suggesting different reward modeling is necessary before tree search can effectively enhance mathematical reasoning in LLMs.
1 Introduction
Mathematical reasoning involves understanding complex problems, decomposing them into manageable steps, and revisiting intermediate results until reaching a sound solution. Large language models (LLMs) have shown remarkable capabilities in solving mathematical problems by breaking solutions into reasoning steps through chain-of-thought (CoT) prompting [1, 2]. When combined with process reward models (PRMs) that evaluate individual reasoning steps, Best-of-N (BoN) identifies the most promising CoT from multiple candidates and has become widely adopted [3, 4, 5, 6]. However, CoTās linear structure fails to capture the branching nature of mathematical reasoning, where multiple strategies are considered, partial arguments explored, and errors necessitate backtracking [7, 8]. Moreover, restricting PRM evaluation to complete CoTs misses opportunities for dynamic guidance.
The tree-of-thought (ToT) framework [9] addresses these limitations by exploring multiple partial reasoning paths and enabling revisions using a reward model to assess the correctness of intermediate solutions. Yet applying ToT with PRMs presents challenges: reasoning trees exhibit intractable branching factors and depth, while PRMs may fail to accurately evaluate intermediate steps [3].
This work proposes an adaptive algorithm to maximize PRM scores over the intractable action space and empirically investigates whether PRM-guided tree search can improve mathematical reasoning in LLMs. We evaluate tree search algorithms under varying PRM quality assumptions against BoN across $23$ diverse mathematical problems, using Qwen2.5-Math-7B-Instruct and its associated PRM as our case study. Key findings reveal that: (1) PRM-guided tree search fails to outperform BoN despite higher costs; (2) Monte Carlo tree search and beam search outperform other PRM-guided tree search methods; (3) PRMs poorly approximate state values and reliability degrades with reasoning depth, suggesting credit assignment issues; and (4) PRMs exhibit limited out-of-distribution generalization. This underperformance stems from tree searchās greater reliance on unreliable PRM scores to guide search, whereas BoN evaluates only complete CoTs. These results highlight the limitations of PRM-guided tree search and BoN, indicating that different reward models may be required for mathematical reasoning.
Limitations.
This work demonstrates that PRM-guided tree search fails to outperform BoN due to PRM limitations: poor reasoning step value estimation, degraded reliability with reasoning depth, and limited out-of-distribution generalization. While our study focuses on a single PRM-model pair, we expect this underperformance to generalize to PRMs exhibiting similar pathologies.
2 Background
2.1 Mathematical reasoning as tree search
We formulate mathematical reasoning as search in a tree -structured Markov decision process $\mathcal{M}=~({\mathbb{S}},{\mathbb{A}},r,t)$ where actions $aā{\mathbb{A}}$ are reasoning steps (text ending with an end-of-reasoning-step token), states $sā{\mathbb{S}}\coloneqq\cup_{i=0}^{T}{\mathbb{A}}^{i}$ are partial reasoning sequences and transitions are deterministic $t(s,a,s^{\prime})={\mathds{1}[s^{\prime}={s\oplus a}]}$ (here ${s\oplus a}$ denotes string concatenation). The root state is the prompt $p$ , and ${\mathbb{T}}$ denotes terminal states containing predictions in the ā \boxed{x} ā format. The reward function assigns $r(s)=1$ if $sā{\mathbb{T}}$ contains the correct solution with valid intermediate reasoning steps, and $r(s)=0$ otherwise. The value function $v(s)=r(s)+\max_{aā{\mathbb{A}}}v({s\oplus a})$ indicates whether any continuation from state $s$ leads to a state with reward $1$ . A LLM $\pi_{\bm{\theta}}$ defines a policy $a\sim\pi_{\bm{\theta}}(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ , while a process reward model $f_{\bm{\phi}}(s)$ estimates $p(v(s)=1\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ . Our goal is finding $s$ with $r(s)=1$ using the LLM and the PRM. Since rewards are unavailable during search, the PRMās approximation quality determines the appropriate search strategy. We distinguish three scenarios:
Scenario 1: PRM as Value Function. If the PRM correctly ranks actions by their true ${p(v(s^{\prime})=1\nonscript\>|\allowbreak\nonscript\>\mathopen{}s^{\prime}={s\oplus a})}$ , optimal search recursively selects $a^{*}=\arg\max_{aā{\mathbb{A}}}f_{\bm{\phi}}({s\oplus a})$ . If actions ${\mathbb{A}}$ were practically enumerable, a greedy tree search algorithm would be optimal. However, enumerating $aā{\mathbb{A}}$ is intractable and we propose an algorithm in Section Ė 3 to adaptively resample actions $a\sim\pi_{\bm{\theta}}(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ until we are confident that we have attained $\max_{aā{\mathbb{A}}}f_{\bm{\phi}}({s\oplus a})$ .
Scenario 2: PRM as Terminal Signal Only. In this āworst-case we can still work withā scenario, the PRM suffers from poor credit assignment, failing to properly estimate $p(v(s)=1\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ for intermediate states, yet PRM scores at terminal states still correlate with reward $r$ . The optimal tree search algorithm returns $s^{*}ā\arg\max_{sā{\mathbb{T}}}f_{\bm{\phi}}(s)$ among terminal states.
Scenario 3: PRM as Noisy Intermediate Signal. The PRM provides useful but unreliable guidance for intermediate states. For example, it may undervalue (i.e., $f_{\bm{\phi}}(s)<p(v(s)=1\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ ) optimal intermediate states leading to high-scoring terminal states, and overvalue (i.e., ${f_{\bm{\phi}}(s)>p(v(s)=1\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}$ ) suboptimal intermediate states leading to poor terminal states. While maximizing PRM scores generally helps to reach valuable terminal nodes, the appropriate search objective remains unclear.
2.2 Tree search baselines
Best-of-N samples $N$ chains-of-thought from the LLM policy (i.e., separate root-to-terminal paths with no shared intermediate state) and selects the CoT with the highest aggregated PRM score [3, 5]. Given a prompt $p$ , we sample CoT $c_{i}=(p,\smash{a^{(i)}_{1}},...,\smash{a^{(i)}_{T}})$ as $\smash{a^{(i)}_{j+1}}\sim\pi_{\bm{\theta}}(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}p\oplus(\oplus_{k=1}^{j}\smash{a^{(i)}_{k}}))$ for $i=1,...,N$ and return $\arg\max_{iā\{1,...,N\}}\Psi(\{f_{\bm{\phi}}(\smash{a_{j}^{(i)}})\}_{j=1}^{T})$ where $\Psi$ aggregates PRM scores.
Greedy best-first search (GBFS) expands the frontier state $sā{\mathbb{F}}$ with highest heuristic value $h(s)$ at each step, where the frontier ${\mathbb{F}}$ contains all unexpanded states in the current search tree. Starting from the root, we repeatedly expand $s=\arg\max_{sā{\mathbb{F}}}h(s)$ by sampling $K$ actions from the LLM until reaching a terminal state. We use $h(s)=f_{\bm{\phi}}(s)$ and depth-aware $h(s)=f_{\bm{\phi}}(s)Ā·(M-d(s))$ to favor deeper states, where $M$ is maximum depth and $d(s)$ is the depth of state $s$ .
Beam search maintains the top- $N$ states with highest (cumulative) PRM score in a beam ${\mathbb{B}}_{t}$ . At each step $t$ , we sample actions $\smash{a_{j}^{i}\sim\pi_{\bm{\theta}}(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}s_{j})}$ $i=1,...,K$ for each state in the current beam $s_{j}ā{\mathbb{B}}_{t}$ , score states $\cup_{j=1}^{N}\{s_{j}\oplus a_{j}^{i}\}_{i=1}^{K}$ with the PRM, and keep the $N$ highest-scoring states for the next beam ${\mathbb{B}}_{t+1}$ . For $N=1$ , this reduces to greedy search.
Monte Carlo tree search (MCTS) builds a search tree in four phases: (1) Select: traverse the search tree from root to leaf selecting high-value states and balancing exploration/exploitation; (2) Expand: add children to the selected leaf by sampling actions from $\pi_{\bm{\theta}}$ ; (3) Rollout: run LLM policy from the new state to a terminal state; (4) Backpropagate: update visit counts and average PRM scores of terminal states along the path back to the root. Repeat until computational budget is depleted.
3 Methods
In this section, we propose an adaptive algorithm to solve the intractable optimization problem $a=\arg\max_{aā{\mathbb{A}}}f_{\bm{\phi}}({s\oplus a})$ (due to the unenumerable $\lvert{\mathbb{A}}\rvert$ ) from Scenario 1 (see Section Ė 2.1). We formulate this optimization problem as a stopping problem: when should we stop sampling actions from the policy and commit to the action with highest observed $f_{\bm{\phi}}({s\oplus a})$ ?
Maximizing over the intractable ${\mathbb{A}}$ using Gittinās indices. At each state $s$ , we sample independent actions $a_{i}\sim\pi_{\bm{\theta}}(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ and evaluate PRM scores $f_{i}=f_{\bm{\phi}}({s\oplus a_{i}})$ . We must then decide whether to stop and commit to the current maximum observed PRM score $m=\max_{i}f_{i}$ (payoff $m$ ), or sample again at cost $c$ to improve the current estimate $m$ for expected payoff $\operatorname{\mathbb{E}}_{f\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}\left[\max(m,f)\right]-c$ (since we select the largest value among $f\sim p(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ and $m$ and incur a cost $c$ ). This is an instance of the Pandoraās box problem [10], whose optimal strategy samples if $\operatorname{\mathbb{E}}_{f\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}\left[\max(m,f)\right]-c>m$ and otherwise stops. This involves computing a Gittinās index $m^{*}$ satisfying $\operatorname{\mathbb{E}}_{f\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}\left[\max(0,f-m^{*})\right]=c$ , then sampling if $m^{*}>m$ and stopping otherwise. However, $p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ is intractable and estimating it requires the LLM samples we are trying to acquire sparingly. This motivates a strategy based on surrogate modeling and posterior inference inspired by Xie et al. [11] which we discuss next.
Bayesian surrogate approximation. We approximate $p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ using a logit-Normal surrogate model $q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,{\bm{\psi}})$ with parameters ${\bm{\psi}}$ . Specifically, we encode prior beliefs in $p({\bm{\psi}})$ , update the posterior $p({\bm{\psi}}\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})\propto q(\mathcal{D}\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,{\bm{\psi}})p({\bm{\psi}})$ with observed PRM scores $\mathcal{D}=\{f_{i}|f_{i}\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)\}$ , and use the posterior predictive $q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,\mathcal{D})=āt q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,{\bm{\psi}})p({\bm{\psi}}\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})d{\bm{\psi}}$ to approximate $p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ . We then compute the Gittinās index $m^{*}$ by solving $\operatorname{\mathbb{E}}_{f\sim q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,\mathcal{D})}\left[\max(0,f-m)\right]=c$ under posterior beliefs $q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,\mathcal{D})$ rather than $p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ . The left-hand side is the expected improvement over $m$ [12], a standard Bayesian optimization acquisition function [13]. The Gittinās index represents the threshold where expected improvement equals cost $c$ , thus smaller $c$ induces more exploration. More details in Appendix Ė A.
4 Results
4.1 Experimental setup
LLM & PRM.
We use the Qwen2.5-Math-7B-Instruct [4] LLM with the recommended prompting strategy and sampling parameters, and the Qwen2.5-Math-PRM-7B process reward model [3].
Problems & metrics.
We evaluate on $22$ mathematical reasoning problems from Yang et al. [4] and AIME 2025 [14]. We report mean accuracy and rank across problems with standard errors. To address concerns about high variance in LLM evaluation [15], we test statistical significance between the top-performing method and all others using Wilcoxon signed-rank tests (insignificant if $p>0.05$ ).
Tree search methods.
We compare Best-of-N (BoN) with $N=8$ chain-of-thoughts using last, minimum, average, product, maximum and sum aggregation functions $\Psi$ ; the proportion of answers containing at least one correct prediction among $N=8$ CoTs (PASS@N); majority voting among predictions of $N=8$ CoTs (MAJ@N); beam search with beam size $N=1$ expanding the state with highest PRM value from $K$ policy samples (Greedy@K); beam search with beam size $N=4$ from $K=6$ policy samples maximizing instantaneous (V) or cumulative (CV) PRM scores (Beam@N); greedy best-first search with $K=8$ policy samples (GBFS@K); depth-aware GBFS (GBFS_DA@K; see Section Ė 2); Monte Carlo tree search with $K=8$ policy samples (MCTS@K) and our proposed method from Section Ė 3 with constant and linear cost schedules to allow more exploration early in the search when the remaining sampling budget is large (Gittins@cost; more details in Appendix Ė A).
4.2 Findings
Table 1: Method mean accuracy and rank with standard errors across problems. We bold results which are not significantly worse than the best ( $p>0.05$ ).
| Method PASS@8 MAJ@8 | Accuracy (p-value) 79.8 $Ā±$ 4.7 (N/A) 0 71.4 $Ā±$ 5.1 (0.010) | Rank (p-value) N/A 4.43 $Ā±$ 0.51 (0.022) |
| --- | --- | --- |
| BoN_Last@8 | 72.7 $Ā±$ 5.1 (N/A) 0 | 3.13 $Ā±$ 0.38 (N/A) 0 |
| BoN_Avg@8 | 72.1 $Ā±$ 5.0 (0.444) | 3.22 $Ā±$ 0.31 (0.787) |
| BoN_Min@8 | 72.2 $Ā±$ 5.0 (0.711) | 3.26 $Ā±$ 0.32 (0.608) |
| BoN_Prod@8 | 72.0 $Ā±$ 5.0 (0.408) | 3.26 $Ā±$ 0.40 (0.795) |
| BoN_Sum@8 | 67.6 $Ā±$ 5.3 (0.000) | 7.30 $Ā±$ 0.72 (0.000) |
| BoN_Max@8 | 68.8 $Ā±$ 5.4 (0.000) | 7.35 $Ā±$ 0.69 (0.000) |
| Greedy@6 | 71.6 $Ā±$ 5.0 (0.043) | 5.74 $Ā±$ 0.76 (0.003) |
| Greedy@20 | 71.2 $Ā±$ 5.0 (0.039) | 5.09 $Ā±$ 0.55 (0.009) |
| Beam@4 (V) | 71.8 $Ā±$ 4.9 (0.126) | 3.83 $Ā±$ 0.42 (0.236) |
| Beam@4 (CV) | 71.9 $Ā±$ 4.8 (0.189) | 4.00 $Ā±$ 0.49 (0.221) |
| GBFS@8 | 46.0 $Ā±$ 4.1 (0.000) | 10.09 $Ā±$ 0.71 (0.000) |
| GBFS_DA@8 | 48.1 $Ā±$ 4.6 (0.000) | 10.00 $Ā±$ 0.65 (0.000) |
| MCTS@8 | 71.2 $Ā±$ 5.0 (0.987) | 3.26 $Ā±$ 0.69 (0.856) |
| Gittins@0.05 (ours) | 70.5 $Ā±$ 5.2 (0.012) | 5.96 $Ā±$ 0.59 (0.001) |
| Gittins@linear (ours) | 71.4 $Ā±$ 5.1 (0.013) | 4.74 $Ā±$ 0.56 (0.011) |
1. PRM-guided tree search methods do not outperform Best-of-N despite higher costs.
Best-of-N using terminal PRM scores (BoN_Last@8) achieves the highest mean rank and accuracy (see Tables Ė 1 and 4). Among Best-of-N variants, average, minimum, and product aggregations perform comparably without significant differences. MCTS and beam search also show no significant performance degradation compared to the best method. However, tree search methods incur substantially higher computational costs, generating considerably more tokens (see Generated token in Table Ė 3). Despite this increased cost, their final solutions contain fewer reasoning steps and tokens than Best-of-N solutions (see Reasoning steps and Out Tokens in Table Ė 3). Except for GBFS, tree search methods reach approximately as many terminal states as Best-of-N (see Terminal states in Table Ė 3).
Table 2: Method mean accuracy and rank with standard errors across problems for tree search methods. We bold results which are not significantly worse than the best ( $p>0.05$ ).
| Method Greedy@6 Greedy@20 | Accuracy (p-value) 71.6 $Ā±$ 5.0 (0.498) 71.2 $Ā±$ 5.0 (0.019) | Rank (p-value) 3.78 $Ā±$ 0.41 (0.021) 3.43 $Ā±$ 0.27 (0.032) |
| --- | --- | --- |
| Beam@4 (V) | 71.8 $Ā±$ 4.9 (0.601) | 2.52 $Ā±$ 0.20 (0.232) |
| Beam@4 (CV) | 71.9 $Ā±$ 4.8 (N/A) 0 | 2.57 $Ā±$ 0.24 (0.286) |
| GBFS@8 | 46.0 $Ā±$ 4.1 (0.000) | 6.70 $Ā±$ 0.34 (0.000) |
| GBFS_DA@8 | 48.1 $Ā±$ 4.6 (0.000) | 6.48 $Ā±$ 0.30 (0.000) |
| MCTS@8 | 71.2 $Ā±$ 5.0 (0.332) | 2.26 $Ā±$ 0.41 (N/A) 0 |
| Gittins@0.05 (ours) | 70.5 $Ā±$ 5.2 (0.019) | 4.00 $Ā±$ 0.35 (0.006) |
| Gittins@linear (ours) | 71.4 $Ā±$ 5.1 (0.398) | 3.09 $Ā±$ 0.30 (0.048) |
2. MCTS and beam search perform best among PRM-guided tree search methods.
MCTS@8 achieves the highest mean rank among tree search methods, with beam search variants (Beam@4 (V) and Beam@4 (CV)) performing comparably without significant differences ( ${p>0.05}$ , see Table Ė 2). For mean accuracy, beam search maximizing the cumulative PRM values performs best, followed by Beam@4 (V), Greedy@6, Gittins@linear and MCTS@8 with no significant performance gaps.
<details>
<summary>x1.png Details</summary>
### Visual Description
## Line Graph: Correlation vs. Reasoning Steps to Terminal State
### Overview
The image depicts a line graph comparing the correlation of three data categories ("All data," "In-distribution," and "Out-of-distribution") across 50 reasoning steps to a terminal state. All three lines exhibit a general downward trend, with convergence toward lower correlation values as reasoning steps increase.
### Components/Axes
- **Y-axis**: "Correlation" (scale: 0.0 to 1.0, linear increments).
- **X-axis**: "Reasoning steps to terminal state" (scale: 0 to 50, linear increments).
- **Legend**: Located in the top-right corner, with three entries:
- Green: "All data"
- Blue: "In-distribution"
- Red: "Out-of-distribution"
### Detailed Analysis
1. **All data (Green)**:
- Starts at ~0.6 correlation at x=0.
- Gradually declines to ~0.2 by x=50.
- Shows minor fluctuations but maintains a steady downward slope.
2. **In-distribution (Blue)**:
- Begins at ~0.8 correlation at x=0.
- Drops sharply to ~0.4 by x=10, then stabilizes around ~0.3ā0.4 until x=30.
- Declines further to ~0.2 by x=50.
3. **Out-of-distribution (Red)**:
- Starts at ~0.4 correlation at x=0.
- Declines steadily to ~0.2 by x=30, with minor oscillations.
- Remains flat at ~0.2 from x=30 to x=50.
### Key Observations
- All three lines converge to a correlation of ~0.2 by x=50, suggesting diminishing performance across all data types at longer reasoning steps.
- "In-distribution" data begins with the highest correlation but experiences the steepest initial decline.
- "Out-of-distribution" data starts with the lowest correlation but follows a similar long-term trend.
- No significant outliers or anomalies are observed; all lines exhibit smooth, continuous trends.
### Interpretation
The graph demonstrates that increased reasoning steps correlate with reduced performance across all data categories. The "In-distribution" data initially outperforms others but degrades more rapidly, while "Out-of-distribution" data maintains a consistently lower baseline. This suggests that longer reasoning chains may disproportionately impact in-distribution data, potentially due to overfitting or complexity mismatches. The convergence at x=50 implies that extended reasoning steps homogenize performance across data types, possibly due to shared failure modes or saturation effects.
</details>
Figure 1: Correlation of prediction correctness with PRM scores. Correlation decreases with increasing distance in reasoning steps from terminal states.
3. The PRM poorly approximates ${p(v(s)=1\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}$ and reliability degrades with reasoning depth, limiting tree search effectiveness.
Methods assuming the PRM accurately estimates the value across all states (Greedy@K and Gittins@cost, Scenario 1 in Section Ė 2.1) perform significantly worse than Best-of-N and other tree search methods (Tables Ė 1 and 2). Increasing policy samples ( $K=6$ to $K=20$ ) or adaptive sampling (Gittins@cost) does not improve performance. This suggests either the LLM policy cannot generate high-scoring states or the PRM cannot identify them. Since PASS@K significantly outperforms Best-of-N, the LLM policy does generate correct solutions but the PRM fails to rank them highly, providing evidence that the PRM poorly approximates $p(v(s)=1\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ .
Further analysis shows that point-biserial correlation of prediction correctness with PRM scores is initially high ( ${ā 0.5}$ ) near terminal states but deteriorates significantly for early reasoning steps ( $ā 0.37$ at $10$ steps from termination; see All data in Figure Ė 1). These findings suggest credit assignment issues in the offline reinforcement learning of the PRM. Moreover, this pattern explains why MCTS@8, which relies exclusively on terminal PRM scores, achieves the best average rank among tree search methods, and Beam@8 performs best in accuracy by tolerating locally suboptimal steps that lead to higher-scoring future states. This suggests that the PRM operates between Scenarios 2 and 3: terminal scores are most reliable, but intermediate scores provide useful yet unreliable guidance.
4. The PRM shows limited out-of-distribution generalization.
Correlation between PRM scores and correctness is consistently higher on in-distribution (ID) problems on which the PRM is trained (GSM8K and MATH) than out-of-distribution (OOD) tasks (others problems; Figures Ė 2 and 5). This generalization gap persists across most reasoning steps: correlation of prediction correctness with PRM scores on ID problems is considerably larger than on OOD tasks until $ā 30$ steps to termination, after which both ID and OOD performance converge to similarly low correlation levels (see ID and OOD in Figure Ė 1). This limited generalization further constrains the practical utility of PRM-guided tree search across diverse mathematical domains.
5 Related work
Tree search with LLM self-evaluation
Several works apply greedy best-first search [9], Monte Carlo tree search [16, 17] and beam search [18] to reasoning using the LLM policy model itself for state evaluation. Unlike these approaches, we use a process reward model trained by offline reinforcement learning on mathematics tasks to guide search.
PRM-guided tree search
Zhang et al. [3] evaluate PRM aggregation methods and greedy search, finding greedy search generally inferior to Best-of-N. Our study extends this analysis with a more comprehensive evaluation, including additional tree search methods (MCTS, GBFS, beam search) and more than three times as many problems ( $23$ vs. $7$ ). Our findings challenge some of theirs results, notably that last-token aggregation outperforms product aggregation on average. More importantly, we provide evidence that PRM reliability degrades with reasoning depth, a key finding that explains why tree search methods fail to outperform Best-of-N.
Tree search for LLM training
Recent work has used MCTS within reinforcement learning pipelines to improve both language models and PRMs. Zhang et al. [19], Wan et al. [20], Xie et al. [21], Guan et al. [22] employ MCTS under PRM supervision to generate high-scoring reasoning chains subsequently used to fine-tune both the policy and PRM through self-training. In contrast, our work focuses on pretrained LLM and PRM models without fine-tuning.
6 Conclusion
We proposed an adaptive algorithm to maximize PRM scores over the intractable action space, and empirically investigated PRM-guided tree search across $23$ mathematical reasoning problems using Qwen2.5-Math-7B-Instruct and its associated PRM. Our findings show that PRM-guided tree search methods fail to outperform Best-of-N despite higher costs, with MCTS and beam search proving most effective among PRM-guided tree search approaches. We identify the underlying causes: PRMs poorly approximate state values, become less reliable with reasoning depth indicating credit assignment issues, and exhibit limited out-of-distribution generalization restricting their broader applicability. Tree searchās underperformance stems from its reliance on these unreliable intermediate-step PRM scores to guide search, whereas BoN evaluates only complete CoTs. These results highlight the limitations of both PRM-guided tree search and BoN, revealing that current PRMs lack sufficient accuracy to guide dynamic mathematical reasoning and suggesting that different reward models may be required.
Acknowledgments and Disclosure of Funding
GP is supported by the Canada CIFAR AI Chair program. The authors are grateful to Johannes Zenn for many insightful discussions that helped shape the direction of this project, and to both Johannes Zenn and Arsen Sheverdin for their careful reading of the manuscript and valuable feedback that improved its clarity and presentation. The computational resources used in this work were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and by companies sponsoring the Vector Institute [https://vectorinstitute.ai/partnerships/.
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Appendix A Additional method details
We here provide additional details on the method presented in Section Ė 3.
Maximizing over the intractable ${\mathbb{A}}$ using Gittinās indices.
At each state $s$ , we sample independent actions from the policy $a\sim\pi_{\bm{\theta}}(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ and evaluate $f=f_{\bm{\phi}}({s\oplus a})$ , which induces the distribution
$$
p({\bm{\mathrm{f}}}=f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)=\sum_{a\in{\mathbb{A}}}\mathds{1}(f=f_{\bm{\phi}}({s\oplus a}))\pi_{\bm{\theta}}(a\nonscript\>|\allowbreak\nonscript\>\mathopen{}s).
$$
After sampling $i$ actions from the policy, let $m_{i}=\max_{jā¤ i}f_{j}$ denote the maximum PRM score observed so far. We face a choice between two actions:
1. Sample a new action $a_{i+1}\sim\pi(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ , observe $f_{i+1}=f_{\bm{\phi}}({s\oplus a_{i+1}})$ , and incur cost $c$
1. Stop and commit to the current best action with score $m_{i}$
The expected payoff for sampling is $\operatorname{\mathbb{E}}_{f\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}\left[\max(m_{i},f)\right]-c$ , while stopping yields payoff $m_{i}$ . The optimal policy samples when the sampling payoff exceeds the stopping payoff:
$$
\displaystyle\operatorname{\mathbb{E}}_{f\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}\left[\max(m_{i},f)\right]-c \displaystyle>m_{i} \displaystyle\Leftrightarrow\quad\operatorname{\mathbb{E}}_{f\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}\left[\max(0,f-m_{i})\right] \displaystyle>c
$$
The Gittins index $m_{i}^{*}$ is defined as the unique solution to
$$
\operatorname{\mathbb{E}}_{f\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)}\left[\max(0,f-m_{i}^{*})\right]=c
$$
The optimal policy samples if $m_{i}^{*}>m_{i}$ and stops otherwise [10]. However, computing this expectation requires knowledge of $p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ which is intractable due to the summation over the action space ${\mathbb{A}}$ (see Equation Ė A.1). Since estimating this distribution would require the very LLM samples we aim to collect sparingly, we develop a Bayesian surrogate approach inspired by Xie et al. [11] which we discuss next.
Bayesian surrogate modeling
We approximate $p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ using a surrogate model $q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,{\bm{\psi}})$ with parameters ${\bm{\psi}}$ . Specifically, we encode our prior beliefs in $p({\bm{\psi}})$ , then update the posterior $p({\bm{\psi}}\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})\propto q(\mathcal{D}\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,{\bm{\psi}})p({\bm{\psi}})$ with observations $\mathcal{D}=\{f_{i}|f_{i}\sim p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)\}$ before using the posterior predictive $q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,\mathcal{D})=āt q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,{\bm{\psi}})p({\bm{\psi}}\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})d{\bm{\psi}}$ to approximate $p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ . We then compute the Gittinās index $m_{i}^{*}$ under posterior beliefs $q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,\mathcal{D})$ rather than $p(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ by solving
$$
\operatorname{\mathbb{E}}_{f\sim q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,\mathcal{D})}\left[\max(0,f-m_{i})\right]=c
$$
The left-hand side of Equation Ė A.5 is the expected improvement over the current maximum $m$ [12], a standard Bayesian optimization acquisition function [13]. The Gittinās index $m^{*}$ represents the threshold where expected improvement equals cost $c$ , thus smaller $c$ induces more exploration. More details in Algorithm Ė 1.
Adaptive cost scheduling.
To balance exploration and exploitation over the search horizon, we employ time-varying costs:
$$
c(n)=c_{1}+(c_{2}-c_{1})\times\nicefrac{{n}}{{B}}
$$
where $c_{1}<c_{2}$ are initial and final costs, $B$ is the total sampling budget, and $n$ is the current sample count. This schedule promotes exploration early during search when the remaining sampling budget is large and exploitation as resources diminish.
Algorithm 1 Adaptive PRM-guided tree search using Gittinās indices and surrogate modeling.
1:
function Gittins ( $f_{\bm{\phi}}$ , $\pi_{\bm{\theta}}$ , ${\mathbb{T}}$ , $s$ , $K$ , $c$ )
2:
repeat
3:
$\mathcal{D}ā\emptyset$
4:
repeat
5:
$\mathcal{D}ā\mathcal{D}\cup\{(s_{i},f_{\bm{\phi}}(s_{i}))\,|\,a_{i}\sim\pi_{\bm{\theta}}(Ā·\nonscript\>|\allowbreak\nonscript\>\mathopen{}s),s_{i}={s\oplus a_{i}}\}_{i=1}^{K}$ $\triangleright$ Sample from LLM
6:
$mā\max_{(s,f)ā\mathcal{D}}f$ $\triangleright$ Update current observed maximum
7:
Compute posterior predictive $q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s,\mathcal{D})$ $\triangleright$ Update posterior beliefs
8:
Compute Gittinās index $m^{*}$ by solving $\operatorname{\mathbb{E}}_{q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})}\left[\max(0,f-m^{*})\right]=c$ using bisection
9:
until $m^{*}>m$
10:
$sā\max_{(s,f)ā\mathcal{D}}f$ $\triangleright$ Update current state
11:
until $sā{\mathbb{T}}$
12:
return $s$
Logit-Normal surrogate model
Let $\mathcal{D}_{n}=\{f_{i}\}_{i=1}^{n}$ where $f_{i}ā[0,1]$ be a collection of observations from $p({\bm{\mathrm{f}}}=f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)$ at a given state $s$ . We consider a logit-Normal likelihood model for our observations i.e.
$$
q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)={\operatorname{\mathcal{N}}\left(\operatorname{logit}(f);\mu,\sigma^{2}\right)}\frac{1}{f(1-f)}
$$
where $\frac{1}{f(1-f)}$ adjusts for the change of variable. We then specify a Normal-inverse-Gamma prior on the likelihood parameters $\mu$ and $\sigma^{2}$ i.e.
$$
q(\mu,\sigma^{2})={\operatorname{\mathcal{N}}\left(m_{0},v_{0}\sigma^{2}\right)}\mathcal{IG}(\alpha_{0},\beta_{0})
$$
The model is conjugate and both the posterior and the predictive posterior are available in closed form. The prior is chosen to yield a maximally uniform predictive prior.
The prior predictive is
$$
q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mu,\sigma^{2})=\int q(r\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mu,\sigma^{2})q(\mu,\sigma^{2})d\mu d\sigma^{2}={\operatorname{\mathcal{T}}_{2a_{0}}\left(f;m_{0},\frac{b_{0}(1+v_{0})}{a_{0}}\right)}\frac{1}{f(1-f)}
$$
The posterior is then
$$
\displaystyle q(\mu,\sigma^{2}\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D}) \displaystyle=\frac{\prod_{i=1}^{n}q(f_{i}\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)q(\mu,\sigma^{2})}{\int\prod_{i=1}^{n}q(f_{i}\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)q(\mu,\sigma^{2})d\mu d\sigma^{2}} \displaystyle=\frac{C\prod_{i=1}^{n}{\operatorname{\mathcal{N}}\left(\operatorname{logit}(f_{i});\mu,\sigma^{2}\right)}q(\mu,\sigma^{2})}{C\int\prod_{i=1}^{n}{\operatorname{\mathcal{N}}\left(\operatorname{logit}(f_{i});\mu,\sigma^{2}\right)}q(\mu,\sigma^{2})d\mu d\sigma^{2}} \displaystyle=\frac{\prod_{i=1}^{n}{\operatorname{\mathcal{N}}\left(\operatorname{logit}(f_{i});\mu,\sigma^{2}\right)}q(\mu,\sigma^{2})}{\int\prod_{i=1}^{n}{\operatorname{\mathcal{N}}\left(\operatorname{logit}(f_{i});\mu,\sigma^{2}\right)}q(\mu,\sigma^{2})d\mu d\sigma^{2}} \displaystyle={\operatorname{\mathcal{N}}\left(m_{n},v_{n}\sigma^{2}\right)}\mathcal{IG}(\alpha_{n},\beta_{n})
$$
where $C=\prod_{i=1}^{n}\frac{1}{f_{i}(1-f_{i})}$ , $v_{n}^{-1}=v_{0}^{-1}+n$ , $m_{n}=v_{n}^{-1}(v_{0}^{-1}m_{0}+\sum_{i=1}^{n}\operatorname{logit}(f_{i}))$ , $a_{n}=a_{0}+n/2$ and $b_{n}=b_{0}+\frac{1}{2}[m_{0}^{2}v_{0}^{-1}+\sum_{i=1}^{n}\operatorname{logit}(f_{i})^{2}-m_{n}^{2}v_{n}^{-1}]$ .
The predictive posterior is
$$
q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})=\int q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mu,\sigma^{2})q(\mu,\sigma^{2}\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})d\mu d\sigma^{2}={\operatorname{\mathcal{T}}_{2a_{n}}\left(f;m_{n},\frac{b_{n}(1+v_{n})}{a_{n}}\right)}\frac{1}{f(1-f)}
$$
Furthermore, the marginal likelihood has an analytical formulation:
$$
\displaystyle q(\mathcal{D}\nonscript\>|\allowbreak\nonscript\>\mathopen{}m_{0},v_{0},a_{0},b_{0}) \displaystyle=\int\prod_{i=1}^{n}q(f_{i}\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)q(\mu,\sigma^{2})d\mu d\sigma^{2} \displaystyle=\left[\prod_{i=1}^{n}\frac{1}{f_{i}(1-f_{i})}\right]\left[\int\prod_{i=1}^{n}q(\operatorname{logit}(f_{i})\nonscript\>|\allowbreak\nonscript\>\mathopen{}s)q(\mu,\sigma^{2})d\mu d\sigma^{2}\right] \displaystyle=C\left[\frac{v_{n}^{1/2}b_{0}^{a_{0}}\Gamma(a_{n})}{v_{0}^{1/2}b_{n}^{a_{n}}\Gamma(a_{0})\pi^{n/2}2^{n}}\right]
$$
where $C=\prod_{i=1}^{n}\frac{1}{f_{i}(1-f_{i})}$ .
Implementation details
We estimate the expectation $\operatorname{\mathbb{E}}_{q}\left[\max(0,f-m)\right]$ using quadrature reparameterizing the integral to logit-space
$$
\displaystyle\operatorname{\mathbb{E}}_{q}\left[\max(0,f-m)\right] \displaystyle=\int_{m}^{1}(f-m)q(f\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})df \displaystyle=\int_{\operatorname{logit}(m)}^{\operatorname{logit}(1)}(\operatorname{logit}^{-1}(l)-m)q(l\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{D})dl \tag{1}
$$
where $l=\operatorname{logit}(f)$ . We solve for the Gittinās index $m^{*}$ using bissection search as done in Xie et al. [11]. Since the observations $f$ take values in $[0,1]$ , we shrink them using $f_{i}=\epsilon+(1-2\epsilon)f_{\bm{\phi}}(s_{i})$ to avoid singularities at $0$ and $1$ .
Appendix B Additional experimental setup details
Models and hyperparameters.
We use Qwen2.5-Math-7B-Instruct [23] as our base language model, where actions $aā\mathcal{A}$ correspond to text sequences terminated by ā \n\n ā. For process reward modeling, we use Qwen2.5-Math-PRM-7B trained on MATH and GSM8K problems [3]. Following the recommended configuration [23], we set temperature=0.7, top_p=0.8, and repetition_penalty=1.05 for text generation.
Implementation.
Gittins indices are computed via bisection search and numerical integration performed using SciPy [24]. Tree structures are managed through NetworkX [25]. Models are served using vLLM [26] with our inference pipeline adapted from vector-inference [27]. Evaluation follows the protocol established by Yang et al. [4] using their official codebase. https://github.com/QwenLM/Qwen2.5-Math
Computational resources.
All experiments are conducted on NVIDIA RTX 6000 and A40 GPUs, with each model deployed on a single GPU.
Appendix C Additional experimental results
Table 3: Performance metrics across 23 problems: mean accuracy with standard-error, mean rank with standard-error, reasoning steps, generated tokens (Gen.), output tokens (Out.), and terminal state coverage. Bold entries indicate no significant difference from the best method ( $p>0.05$ ). Best-of-N with terminal PRM scores (BoN_Last@8) perform best, while tree search methods (MCTS, beam search) match accuracy but require substantially higher computational cost.
| Method PASS@8 MAJ@8 | Accuracy 79.8 $Ā±$ 4.7 (N/A) 0 71.4 $Ā±$ 5.1 (0.010) | Rank N/A 4.43 $Ā±$ 0.51 (0.022) | Reasoning steps 9.7 9.7 | Gen. Tokens ( $Ć 10^{7}$ ) 8.2 8.2 | Out. Tokens ( $Ć 10^{7}$ ) 8.2 8.2 | Terminal State (%) 98.2 98.2 |
| --- | --- | --- | --- | --- | --- | --- |
| BoN_Last@8 | 72.7 $Ā±$ 5.1 (N/A) 0 | 3.13 $Ā±$ 0.38 (N/A) 0 | 9.7 | 8.2 | 8.2 | 98.2 |
| BoN_Avg@8 | 72.1 $Ā±$ 5.0 (0.444) | 3.22 $Ā±$ 0.31 (0.787) | 9.7 | 8.2 | 8.2 | 98.2 |
| BoN_Min@8 | 72.2 $Ā±$ 5.0 (0.711) | 3.26 $Ā±$ 0.32 (0.608) | 9.7 | 8.2 | 8.2 | 98.2 |
| BoN_Prod@8 | 72.0 $Ā±$ 5.0 (0.408) | 3.26 $Ā±$ 0.40 (0.795) | 9.7 | 8.2 | 8.2 | 98.2 |
| BoN_Sum@8 | 67.6 $Ā±$ 5.3 (0.000) | 7.30 $Ā±$ 0.72 (0.000) | 9.7 | 8.2 | 8.2 | 98.2 |
| BoN_Max@8 | 68.8 $Ā±$ 5.4 (0.000) | 7.35 $Ā±$ 0.69 (0.000) | 9.7 | 8.2 | 8.2 | 98.2 |
| Greedy@6 | 71.6 $Ā±$ 5.0 (0.043) | 5.74 $Ā±$ 0.76 (0.003) | 8.9 | 5.6 | 0.9 | 97.9 |
| Greedy@20 | 71.2 $Ā±$ 5.0 (0.039) | 5.09 $Ā±$ 0.55 (0.009) | 8.8 | 18.7 | 0.9 | 98.2 |
| Beam@4 (V) | 71.8 $Ā±$ 4.9 (0.126) | 3.83 $Ā±$ 0.42 (0.236) | 9.2 | 20.2 | 0.9 | 98.7 |
| Beam@4 (CV) | 71.9 $Ā±$ 4.8 (0.189) | 4.00 $Ā±$ 0.49 (0.221) | 9.4 | 21.0 | 0.9 | 96.5 |
| GBFS@8 | 46.0 $Ā±$ 4.1 (0.000) | 10.09 $Ā±$ 0.71 (0.000) | 5.3 | 65.2 | 0.5 | 52.7 |
| GBFS_DA@8 | 48.1 $Ā±$ 4.6 (0.000) | 10.00 $Ā±$ 0.65 (0.000) | 5.8 | 68.0 | 0.5 | 57.1 |
| MCTS | 71.2 $Ā±$ 5.0 (0.987) | 3.26 $Ā±$ 0.69 (0.856) | 8.8 | 89.4 | 0.7 | 97.8 |
| Gittins@0.05 (ours) | 70.5 $Ā±$ 5.2 (0.012) | 5.96 $Ā±$ 0.59 (0.001) | 9.5 | 9.0 | 0.9 | 98.3 |
| Gittins@linear (ours) | 71.4 $Ā±$ 5.1 (0.013) | 4.74 $Ā±$ 0.56 (0.011) | 9.5 | 14.3 | 0.9 | 98.5 |
Table 4: Accuracy by dataset with means and standard errors. Bold indicates best performance per dataset. Bottom rows show overall means and win counts across problems.
| aime25 aime24 amc23 | 20.0 23.3 82.5 | 13.3 16.7 60.0 | 13.3 16.7 70.0 | 16.7 20.0 67.5 | 16.7 20.0 70.0 | 20.0 20.0 70.0 | 13.3 16.7 62.5 | 6.7 10.0 62.5 | 16.7 16.7 62.5 | 10.0 26.7 62.5 | 13.3 16.7 55.0 | 16.7 13.3 70.0 | 20.0 16.7 67.5 | 16.7 30.0 65.0 | 6.7 6.7 40.0 | 3.3 13.3 37.5 | 23.3 26.7 65.0 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| sat_math | 100.0 | 96.9 | 100.0 | 100.0 | 100.0 | 100.0 | 93.8 | 93.8 | 100.0 | 96.9 | 100.0 | 96.9 | 96.9 | 96.9 | 62.5 | 75.0 | 96.9 |
| aqua | 94.1 | 78.0 | 76.0 | 76.0 | 75.6 | 76.0 | 73.2 | 74.4 | 79.5 | 79.5 | 78.0 | 75.5 | 77.6 | 79.5 | 63.8 | 63.0 | 68.3 |
| asdiv | 96.7 | 95.8 | 96.0 | 96.0 | 95.9 | 96.0 | 95.9 | 95.5 | 96.1 | 96.0 | 95.8 | 95.8 | 96.0 | 95.8 | 70.7 | 76.1 | 96.0 |
| carp_en | 63.1 | 61.8 | 61.7 | 61.9 | 62.0 | 62.0 | 61.7 | 61.5 | 61.0 | 61.2 | 61.1 | 61.1 | 61.1 | 61.5 | 36.1 | 39.2 | 61.5 |
| cmath | 95.3 | 92.7 | 93.2 | 93.8 | 94.0 | 94.0 | 92.2 | 92.2 | 92.3 | 93.2 | 92.8 | 93.3 | 93.0 | 93.5 | 69.8 | 73.5 | 94.0 |
| cn_middle_school | 82.2 | 78.2 | 80.2 | 79.2 | 79.2 | 79.2 | 76.2 | 76.2 | 80.2 | 78.2 | 77.2 | 77.2 | 77.2 | 78.2 | 54.5 | 61.4 | 79.2 |
| gaokao_math_cloze | 81.4 | 76.3 | 78.0 | 78.0 | 78.0 | 78.0 | 72.9 | 76.3 | 78.0 | 76.3 | 76.3 | 79.7 | 78.8 | 77.1 | 47.5 | 57.6 | 75.4 |
| gaokao_math_qa | 94.6 | 73.5 | 75.8 | 77.8 | 77.2 | 77.5 | 56.1 | 73.5 | 68.4 | 70.4 | 72.9 | 72.9 | 70.9 | 63.8 | 52.1 | 56.4 | 75.5 |
| gaokao2023en | 80.8 | 72.2 | 73.0 | 71.9 | 72.2 | 71.4 | 69.9 | 67.0 | 69.4 | 69.6 | 69.6 | 71.7 | 71.7 | 70.1 | 36.6 | 40.5 | 75.6 |
| gaokao2024_I | 71.4 | 57.1 | 57.1 | 50.0 | 57.1 | 50.0 | 42.9 | 57.1 | 64.3 | 64.3 | 64.3 | 57.1 | 57.1 | 57.1 | 42.9 | 35.7 | 64.3 |
| gaokao2024_II | 85.7 | 64.3 | 71.4 | 64.3 | 57.1 | 57.1 | 50.0 | 50.0 | 71.4 | 57.1 | 57.1 | 64.3 | 64.3 | 64.3 | 35.7 | 28.6 | 57.1 |
| gaokao2024_mix | 79.1 | 72.5 | 73.6 | 71.4 | 71.4 | 71.4 | 59.3 | 71.4 | 65.9 | 68.1 | 64.8 | 67.0 | 68.1 | 69.2 | 46.2 | 36.3 | 79.4 |
| gsm8k | 97.7 | 96.6 | 96.4 | 96.4 | 96.4 | 96.4 | 96.1 | 95.7 | 96.1 | 96.0 | 95.6 | 96.0 | 96.7 | 96.6 | 46.3 | 54.4 | 96.7 |
| mawps | 98.8 | 98.5 | 98.4 | 98.5 | 98.5 | 98.5 | 98.4 | 98.3 | 98.1 | 98.3 | 98.4 | 98.4 | 98.5 | 98.3 | 72.5 | 81.7 | 98.4 |
| minerva_math | 48.9 | 41.5 | 39.3 | 40.1 | 40.1 | 39.3 | 37.1 | 36.0 | 38.2 | 38.2 | 39.3 | 37.9 | 40.8 | 40.4 | 14.0 | 15.1 | 42.1 |
| mmlu_stem | 90.2 | 72.9 | 74.0 | 73.4 | 73.7 | 72.9 | 69.0 | 70.2 | 72.0 | 72.4 | 72.2 | 74.0 | 73.6 | 73.5 | 54.6 | 43.0 | 31.7 |
| svamp | 96.7 | 94.6 | 95.0 | 95.1 | 95.2 | 95.1 | 94.2 | 93.4 | 94.9 | 95.6 | 95.5 | 95.9 | 95.6 | 95.7 | 68.7 | 76.4 | 96.8 |
| tabmwp | 98.6 | 96.1 | 96.4 | 96.5 | 96.4 | 96.9 | 95.0 | 94.6 | 95.5 | 96.0 | 95.3 | 96.1 | 96.6 | 96.7 | 63.3 | 68.3 | 96.2 |
| olympiadbench | 61.9 | 44.7 | 48.4 | 47.3 | 46.1 | 46.7 | 42.1 | 42.7 | 44.1 | 45.5 | 43.3 | 44.9 | 46.1 | 46.1 | 21.6 | 21.9 | 47.9 |
| math | 92.3 | 87.0 | 88.2 | 87.6 | 87.5 | 87.3 | 85.5 | 84.5 | 86.0 | 86.4 | 86.7 | 86.4 | 86.7 | 86.8 | 44.7 | 48.8 | 89.1 |
| Average | 79.8 $Ā±$ 4.7 | 71.4 $Ā±$ 5.1 | 72.7 $Ā±$ 5.1 | 72.1 $Ā±$ 5.0 | 72.2 $Ā±$ 5.0 | 72.0 $Ā±$ 5.0 | 67.6 $Ā±$ 5.3 | 68.8 $Ā±$ 5.4 | 71.6 $Ā±$ 5.0 | 71.2 $Ā±$ 5.0 | 70.5 $Ā±$ 5.2 | 71.4 $Ā±$ 5.1 | 71.8 $Ā±$ 4.9 | 71.9 $Ā±$ 4.8 | 46.0 $Ā±$ 4.1 | 48.1 $Ā±$ 4.6 | 71.2 $Ā±$ 5.0 |
| Best Count | N/A | 1/23 | 5/23 | 2/23 | 4/23 | 5/23 | 0/23 | 0/23 | 4/23 | 0/23 | 1/23 | 3/23 | 1/23 | 1/23 | 0/23 | 0/23 | 6/23 |
<details>
<summary>x2.png Details</summary>
### Visual Description
## Bar Chart: Correlation Metrics Across Data Types
### Overview
The chart compares correlation values across six metrics (Last, Avg, Min, Prod, Sum, Max) for three data types: All data (green), In-distribution (blue), and Out-of-distribution (red). The y-axis represents correlation strength (0.0ā1.0), with values labeled on top of each bar.
### Components/Axes
- **X-axis (Metrics)**: Last, Avg, Min, Prod, Sum, Max
- **Y-axis (Correlation)**: 0.0 to 1.0 in increments of 0.1
- **Legend**:
- Green = All data
- Blue = In-distribution
- Red = Out-of-distribution
- **Bar Structure**: Each metric has three grouped bars (green, blue, red) aligned vertically.
### Detailed Analysis
1. **Last**:
- All data: 0.56
- In-distribution: 0.70
- Out-of-distribution: 0.50
2. **Avg**:
- All data: 0.55
- In-distribution: 0.72
- Out-of-distribution: 0.49
3. **Min**:
- All data: 0.57
- In-distribution: 0.70
- Out-of-distribution: 0.53
4. **Prod**:
- All data: 0.53
- In-distribution: 0.59
- Out-of-distribution: 0.52
5. **Sum**:
- All data: 0.02
- In-distribution: 0.09
- Out-of-distribution: -0.02
6. **Max**:
- All data: 0.21
- In-distribution: 0.20
- Out-of-distribution: 0.22
### Key Observations
- **In-distribution data** consistently shows the highest correlation across all metrics except "Max," where it is slightly lower than All data.
- **Out-of-distribution data** exhibits the weakest correlations, with a notable negative value (-0.02) in the "Sum" metric.
- **All data** performs intermediately, often closer to Out-of-distribution than In-distribution.
- The "Sum" metric is an outlier, with all data types showing near-zero or negative correlations.
### Interpretation
The data suggests that **In-distribution samples** maintain stronger relationships with the target variable compared to All data and Out-of-distribution samples. The negative correlation in "Sum" for Out-of-distribution data implies an inverse relationship, potentially indicating data quality issues or domain shifts. The "Max" metricās low correlations across all types may reflect diminishing returns or saturation effects at extreme values. This pattern highlights the importance of domain alignment in predictive modeling, as Out-of-distribution data degrades performance significantly.
</details>
Figure 2: Point-biserial correlation between PRM score aggregation methods and solution correctness. Minimum aggregation shows highest correlation, followed by last reasoning-step scoring. Correlation is consistently higher on in-distribution (ID) problems on which the PRM is trained (GSM8K and MATH) than out-of-distribution (OOD) tasks.
<details>
<summary>x3.png Details</summary>
### Visual Description
## Bar Chart: PRM Score Distributions by Correctness
### Overview
The image contains six bar charts comparing distributions of PRM (Problem Resolution Metric) scores between correct and incorrect answers. Each chart uses a density scale (y-axis) and a normalized score range (x-axis: 0.0ā1.0 or 0ā500). Blue bars represent correct answers, red bars represent incorrect answers. All charts share consistent axis labeling and legend placement.
### Components/Axes
1. **X-Axes**:
- Top row: "Last PRM score", "Average of PRM scores", "Minimum of PRM scores"
- Bottom row: "Product of PRM scores", "Sum of PRM scores", "Maximum of PRM scores"
2. **Y-Axes**: "Density" (scale varies per chart, max ~20)
3. **Legend**: Located in bottom-right corner, blue = Correct, red = Incorrect
4. **Chart Layout**: Two rows of three charts each, uniform styling
### Detailed Analysis
1. **Last PRM Score**
- Correct: Single blue bar at x=1.0 (density ~20)
- Incorrect: Single red bar at x=0.0 (density ~5)
- *Trend*: Perfect separation between groups
2. **Average PRM Scores**
- Correct: Blue bar at x=1.0 (density ~20)
- Incorrect: Red bar at x=0.0 (density ~5)
- *Trend*: Identical to "Last PRM Score" chart
3. **Minimum PRM Scores**
- Correct: Blue bar at x=1.0 (density ~20)
- Incorrect: Red bar at x=0.0 (density ~5)
- *Trend*: Consistent binary distribution
4. **Product of PRM Scores**
- Correct: Blue bar at x=1.0 (density ~20)
- Incorrect: Red bar at x=0.0 (density ~5)
- *Trend*: Perfect separation maintained
5. **Sum of PRM Scores**
- Correct: Blue bar at x=500 (density ~20)
- Incorrect: Red bar at x=0 (density ~5)
- *Trend*: Absolute separation at extremes
6. **Maximum PRM Scores**
- Correct: Blue bar at x=1.0 (density ~20)
- Incorrect: Red bar at x=0.0 (density ~5)
- *Trend*: Binary distribution pattern
### Key Observations
1. **Binary Performance**: All metrics show perfect separation between correct (1.0/500) and incorrect (0.0/0) scores
2. **Density Consistency**: Correct answers consistently show higher density (~20 vs ~5)
3. **Normalization**: Top charts use 0.0ā1.0 scale, bottom charts use absolute values (0ā500)
4. **Legend Placement**: Bottom-right corner, clearly labeled with color coding
### Interpretation
The data demonstrates a binary outcome pattern where correct answers achieve maximum performance across all metrics, while incorrect answers show zero performance. This suggests:
1. **Threshold Effect**: A clear pass/fail distinction exists in the dataset
2. **Metric Correlation**: All PRM-derived metrics (last, average, min, product, sum, max) align perfectly in separating correct/incorrect answers
3. **Sum Significance**: The 500-point maximum in the sum chart likely represents total possible points, with correct answers achieving full marks
4. **Systematic Errors**: Incorrect answers consistently fail across all evaluation dimensions
The uniformity of results across metrics indicates either:
- A highly reliable scoring system with no false positives/negatives
- Potential overfitting in the evaluation methodology
- A dataset with extreme performance separation (e.g., 100% correct vs 0% incorrect)
This pattern warrants investigation into whether the metrics are capturing meaningful distinctions or reflecting artificial boundaries in the scoring system.
</details>
Figure 3: PRM score distributions by aggregation method, conditioned on prediction correctness. Last, average, minimum, and product aggregations effectively separate correct from incorrect predictions.
Table 5: Point-biserial correlations between PRM aggregation methods and prediction correctness by dataset. Bold indicates best performance per dataset. Correlations vary substantially across problems, with optimal aggregation being dataset-dependent. Overall, minimum aggregation performs best, followed by last-step scoring.
| aime25 aime24 amc23 | 0.494 0.623 0.668 | 0.474 0.597 0.700 | 0.582 0.680 0.726 | 0.444 0.487 0.626 | 0.283 0.529 0.170 | 0.095 0.151 0.280 |
| --- | --- | --- | --- | --- | --- | --- |
| sat_math | 0.690 | 0.375 | 0.487 | 0.330 | 0.069 | 0.039 |
| aqua | 0.309 | 0.309 | 0.342 | 0.306 | 0.106 | 0.291 |
| asdiv | 0.447 | 0.466 | 0.450 | 0.424 | -0.049 | 0.116 |
| carp_en | 0.137 | 0.149 | 0.154 | 0.150 | -0.026 | 0.060 |
| cmath | 0.557 | 0.559 | 0.627 | 0.637 | 0.063 | 0.200 |
| cn_middle_school | 0.432 | 0.419 | 0.465 | 0.444 | -0.161 | 0.119 |
| gaokao_math_cloze | 0.455 | 0.500 | 0.555 | 0.482 | 0.094 | 0.098 |
| gaokao_math_qa | 0.510 | 0.454 | 0.546 | 0.488 | -0.016 | 0.216 |
| gaokao2023en | 0.561 | 0.590 | 0.602 | 0.540 | 0.148 | 0.175 |
| gaokao2024_I | 0.285 | 0.275 | 0.438 | 0.337 | -0.064 | 0.126 |
| gaokao2024_II | 0.376 | 0.447 | 0.545 | 0.584 | 0.048 | 0.094 |
| gaokao2024_mix | 0.482 | 0.422 | 0.495 | 0.343 | 0.081 | 0.187 |
| gsm8k | 0.560 | 0.595 | 0.597 | 0.565 | 0.099 | 0.120 |
| mawps | 0.333 | 0.379 | 0.322 | 0.296 | 0.015 | 0.091 |
| minerva_math | 0.349 | 0.416 | 0.477 | 0.474 | -0.022 | 0.183 |
| mmlu_stem | 0.360 | 0.280 | 0.281 | 0.266 | 0.085 | 0.182 |
| svamp | 0.700 | 0.706 | 0.694 | 0.673 | 0.109 | 0.239 |
| tabmwp | 0.542 | 0.591 | 0.588 | 0.570 | 0.090 | 0.211 |
| olympiadbench | 0.571 | 0.603 | 0.635 | 0.578 | 0.213 | 0.145 |
| math | 0.708 | 0.718 | 0.703 | 0.585 | 0.130 | 0.200 |
| Best Count | 2/23 | 5/23 | 14/23 | 2/23 | 0/23 | 0/23 |
| Average | 0.557 | 0.550 | 0.575 | 0.533 | 0.024 | 0.209 |