ReST-MCTS*: LLM Self-Training via Process Reward Guided Tree Search
Authors: Dan Zhang
Sining Zhoubian
Ziniu Hu
Yisong Yue
Yuxiao Dong
Jie Tang
Venue: NeurIPS 2024 poster (2024)
Keywords: LLM self-training
process reward model
tree search
Primary area: natural_language_processing

Abstract: Recent methodologies in LLM self-training mostly rely on LLM generating responses and filtering those with correct output answers as training data. This approach often yields a low-quality fine-tuning training set (e.g., incorrect plans or intermediate reasoning). In this paper, we develop a reinforced self-training approach, called ReST-MCTS*, based on integrating process reward guidance with tree search MCTS* for collecting higher-quality reasoning traces as well as per-step value to train policy and reward models. ReST-MCTS* circumvents the per-step manual annotation typically used to train process rewards by tree-search-based reinforcement learning: Given oracle final correct answers, ReST-MCTS* is able to infer the correct process rewards by estimating the probability this step can help lead to the correct answer. These inferred rewards serve dual purposes: they act as value targets for further refining the process reward model and also facilitate the selection of high-quality traces for policy model self-training. We first show that the tree-search policy in ReST-MCTS* achieves higher accuracy compared with prior LLM reasoning baselines such as Best-of-N and Tree-of-Thought, within the same search budget. We then show that by using traces searched by this tree-search policy as training data, we can continuously enhance the three language models for multiple iterations, and outperform other self-training algorithms such as ReST$^\text{EM}$ and Self-Rewarding LM.

ReST-MCTS. LLM Self-Training via Process Reward Guided Tree Search
Dan Zhang l+t Sining Zhoubian', Ziniu Hu?* Yisong Yue?, Yuxiao Dong', Jie Tang' The Knowledge Engineering Group (KEG), Tsinghua University;
CCliifornia Institute of Technology
(zd21,zbsn22 Jemails, tsinghua. edu.
https://rest-met thub. io/
Abstract
Recent methodologies in LLM self-training mostly rely LLM generating sponses and filtering those with correct output answers as training data. This approach often yields low-quality fine-tuning training set (e.g., incorrect plans or intermediate reasoning). In this paper, we develop reinforced self-training approach, called ReST-MCTS', based on integrating process reward guidance with tree search MCTS for collecting higher-quality reasoning traces as well as per-step value to train policy and reward models. ReST-MCTS' cireumvents the per-step manual annotation typically used to train process rewards by seee-sarcc--bacee reinforcement learning: Given oracle final correct answers, ReST-MCTS able infer the correct process rewards by estimating the probability this step can help lead the correct answer. These inferred rewards serve dual purposes: they act as value targets for further refining the process reward model and also facilitate the selection of high-quality traces for policy model self-training. We first show that the tree- search policy in ReST-MCTS' achieves higher accuracy compared with prior LLM reasoning baselines such as Best-of-N and Tree-of-Thought, within the same search budget. We then show that by using traces searched by this tree-search policy training data, we can continuously enhance the three language models for multiple iterations, and outperform other self- training algorithms such as RcSTEM and Self- Rewarding LMM. We release all code at https: : //gi thub. www/ TTTHH//U/RRDMMTTTTT
1 Introduction
Large Language Models (LLMs) are mostly trained on human- gennrated data. But as we approach the point where most available high-quality human- human-produced produced text on the web has been crawled and used for I.LM training [1], the research focus has shifted towards using L.MM J.M--gnnrated content to conduct self- self-training [2: 3; 4; 5; 6; 7J. Similar to most Reinforcement Learning (RL) problems, LLM self-training requires a reward signal. Most existing reinforced self-improvement approaches (e.g., STaR [4], RFT [5], RM [6]. V-STaR [7]) assume have access ground-truth reward model (labels from supervised dataset, or a pre-trained reward model). These approaches use an LLM to generate multiple samples for each question, and assume the one that leads to high reward (correct solution) is the high-quality sample, and later train on these samples (hence self-training). Such procedures can be effective in improving LLM performance, in some cases solving reasoning tasks that the base LLM cannot otherwise solve [8; 9; 10].
However, a key limitation of the above procedure is that even if a reasoning trace results in correct solution, it does not necessarily imply that the entire trace is accurate. L.LMs often generate
'Equal contribution.
Work done while DZ visited at Caltech.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Table Key differences between existing self self-improvement methods and our approach. Train refers to whether to train reward model.
Mothod
Reasoning Policy
Reward Guidance
Method
Value Label
Train
STaR H1
CoI+Rellexion
ReST /6] RFT RPO [II]
CoT
Final outcome reward annotated by ground- -ooth answer
Verify Step-hy-Step [1]
Best-of-N
Per-stop process reward innotated by human
MATH--HEPPEERE (121/ /121/PDPO//1 [13]
Best-of-N
Per-step process reward inferred from rundom rollout
TS-LLM [14]
MCTS
Per-stop process reward inferred from TD-A [15]
V-STaR [7]
CoT
Final outcome reward generated by multi-itenation 1.I Ms
Self- Self-Rewarding [16]
CoT
Final outcome reward generated and judged by L.LMs
ReST-MCTS' (Ouns)
MCTS
Per-step process reward inferred from tree search (MCTS') Multi her
wrong useless intermediate reasoning steps, while still finding the correct solution by chance [17]. Consequently, a self-training dataset can often contain many false positives - intermediate reasoning traces plans are incorrect, but the final output is correct- which limits the final performance of LLM fine-tuning for complex reasoning tasks [18; 19]. One way to tackle this issue is to use a value function reward model to verify reasoning traces for correctness (which then serves learning signal for self-training! [1; 12]. However, training a reliable reward model to verify every step in reasoning trace generally depends on dense numan-generated annotations (per reasoning step) [1], which does not scale well. Our research aims to address this gap by developing a novel approach that automates the acquisition of reliable reasoning traces while effectively utilizing reward signals for verification purposes. Our key research question is: How can we automatically acquire high-quality reasoning traces and effectively process reward signals for verification and LLM sell-training?
In this paper, we propose ReST-MCTS", framework for training LLMs using model- RL training. Our proposed approach utilizes a modified Monte Carlo Tree Search (MCTS) algorithm as the reasoning policy, denoted MCTS*, guided by trained per-step process reward (value) model. A key aspect of our method is being able to automatically generate per-step labels for training per-step reward models, by performing sufficient number of rollouts. This labeling process effectively filters out the subset of samples with the highest quality, without requiring additional human intervention. Table 1 summarizes the key distinctions between approach and previous approaches. We validate experimentally that ReST-MCTS* outperforms prior work in discovering good reasoning traces, such as CelffConsistency (SC) and Best-of-N (BoN) under the same search budget on the SciBench [20] and MATH [21] benchmarks, which consequently leads to improved self-training.
To summarize, our contributions are:
We propose ReST-MCTS'. self-training approach that generates process rewards searched by MCTS A key step is to automatically annotate the process reward of each intermediate node via sufficient times of rollouts, using MCTS* We validate multiple reasoning benchmarks and find that ReST-MCTS' outperforms existing self-training approaches (e.g., ReSTEM and Self-Rewardih as shown Table 2 and reasoning policies (e.g., CoT and ToT) as shown in Table 4.
The reward generator in ReST-MCTS' leads to a higher higher-quality process reward model compared to previous process reward generation techniques, e.g., MATH- SHEPHRD, as shown in Table 3.
Given the same search budget, the search algorithm (MCTS') in ReST-MCTS* achieves higher accuracy than Self-Consistency and Best-of-N, as shown in Figure 2.
2 Background on Reasoning & Self-Training
We follow the standard setup in LLM based reasoning. We start with policy, denoted by x, that instantiated using base LLM. Given an input problem Q. in the simplest case, π can generate an output sequence, or trace, of reasoning steps ($1, $2, .K) ~ TT-IQ) by autoregressively predicting the next token. For simplicity, we assume a reasoning step comprises a single sentence (which itself comprises multiple tokens). We also assume the last output 8 is the final step. LLMs can also be prompted or conditioned to bias the generation along certain traces. For a prompt e, we can write the policy as TT-1Q,)) c). This idea was most famously used in tf-ouughht (CoT) [22].

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Self- Seff-Consistency (SC). Self-Consistence [23] samples multiple reasoning traces from π and chooses the final answer that appears most frequently.
Tree-Search & Value Function. Another idea is to use tree-structured reasoning traces [24; 14], that branch from intermediate reasoning steps. One key issue in using so-called tree-search reasoning algorithm is the need to have a value function to guide the otherwise combinatorially large search process [[4]. Two common value functions include Outcome Reward Models (ORMs) [25]. which are trained only on the correctness of the final answer, and Process Reward Models (PRMs) [1], which are trained on the correctness of each reasoning step. We assume Ts, is the PRM's output sigmoid score at k-th step. Our ReST-MCTS' approach uses tree-search to automatically learn good PRM.
Best-of-N. As an alternative to Seff-Consistency, one can also use a learned value function (PRM or ORM) to select the reasoning trace with the highest value [1].
Self-Training. At a high level, there are two steps to self self-training [6; 12]. The first step is generation, where we sample multiple reasoning traces using (in our case, tree-structured traces). The second step is improvement, where a learning signal is constructed on the reasoning traces, which is then used to fine-tune т.. The process can repeat for multiple iterations.
Limitation of Prior Works. The main challenge in doing reliable self faff-training the construction of a useful learning signal. Ideally, one would want dense learning signal on the correctness of every intermediate reasoning step, which is given by PRM. Otherwise, with sparse learning signals, one suffers from a credit assignment similar to that in reinforcement learning. Historically, the main challenge with learning PRM is the lack of supervised annotations per reasoning step. This is the principal challenge that our ReST-MCTS* approach seeks to overcome. We describe detailed preliminaries Appendix A.
3
The ReST-MCTS" Method
Our approach, ReST-MCTS", is outlined in Figure and developed using four main components.
MCTS which performs tree search with sufficient rollout time under the guidance of the PRM.
Process Reward Model (PRM) which evaluates any partial solution's 's quality and guides MCTS*
Policy Model which generates multiple intermediate reasoning steps for each question.
LLM Seainng,, which uses MCTS* to collect reasoning traces, trains policy model on positive samples, and trains process reward model on all generated traces.
3.1 Search-based Reasoning Policy for LLM
Value eA for a Partial Solution. The value (process) reward of the partial solution [81, "., 8k] should satisfy the following basic qualities:
Limited range: Us is constrained within a specific range. This restriction ensures that the values of U are bounded and do not exceed certain limit.
Reflecting probability of correctness: Uj reflects the probability that partial solution is complete and correct answer. Higher values of U) indicate better quality or a higher likelihood of being closer to a correct answer.
Reflecting correctness and contribution of solution steps: Uy incorporates both the correctness and contribution of each solution step. When starting from partial solution, correct next step should result in a higher 03 compared to false ones. Additionally, step that makes more correct deductions toward the final answer should lead to a higher es value. This property ensures that Uk captures the incremental progress made towards the correct solution and rewards steps that contribute to the overall correctness of the solution.
Reasoning Distance my for a Partial Solution. To estimate the progress of a solution step, we define the reasoning distance my of pk as the minimum reasoning steps policy model requires to reach the correct answer, starting from Ps- Reasoning distance reflects the progress made as well as the difficulty for a policy figure out correct answer based current steps, thus it can be further

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MCTS
(0:.)))
S1
V 0
al. 10:)
W 0.25
V
P 0.25
$2 ~ r(. (Q),s))
0.25
Re. (ueeet ((Q.))
legn Fel
53 ~ T. (2)
3,1
0.5
3.N
LLM Self-Training
0.25
PRM
Policy Model
$4.1
0.75
0.25 W
D5o
Dso
M: number of prompts number samples
1
A
==1
A2
1
(iiser vi)
(2. = (Qz. T2
(RiSer, v
j: j-th itcration
A1 V
A # Y1
T: length of'a trace
(51.521'))))
Figure I: The left part presents the process of inferring process rewards and how we conduct process reward guide tree-search. The right part denotes the self-training of both the process reward model and the policy model.
used evaluate the quality of pt- However, we point out that my can not be directly calculated. more like hidden variable that can be estimated by performing simulations trace sampling starting from pk and finding the actual minimum steps used to discover the correct answer.
Weighted Reward Wx, for a Single Step. Based on the desired qualities for evaluating partial solutions, we introduce the concept weighted reward to reflect the quality of the current step denoted as w. Based on the common PRM reward T... Mss, further incorporates the reasoning distance mg as a weight factor, reflecting the incremental progress S3 makes.
Representations for Quality Value and Weighted Reward. To determine the quality value ty of a partial solution at step k. incorporate the previous quality value and the weighted reward of the current step. By considering the previous quality value, we account for the cumulative progress and correctness achieved up to the preceding step. Therefore, the Us can be iteratively updated as:
k ==0
0k maz(vk-1- Wog 0).
else
The weighted reward Way of the current step provides a measure of the quality and contribution of that specific step towards the overall solution. Based on my (where m, = K - and K is the total number of reasoning steps of solution s), previous quality value Uk-1, and res in MATH-SHEPHER we can update the definition of the weighted reward Was iteratively as follows:
[12],
1 Uk-1 2r k 1,2,
Waa
Vilk +
As k increases, my decreases, indicating that fewer reasoning steps are needed to reach the correct answer. This leads to higher weight placed on the weighted reward of the current step. We can also derive that Was and U3 satisfy the expected boundedness shown in the theorem below.
Theorem (Boundedness of Ws, and Ug). If is a sigmoid score ranged between [o, 1], then wsy and uk defined above satisfy following boundedness: W.s. S Uk-1, Uk E [0, 1).
Derivation. Please refer to the detailed derivation in Appendix B.
Therefore, conclude that W.s and e2 has following properties that match expectations:
If reasoning route starting from pk requires more steps to get to the correct answer, then the single-step weighted reward Wsy is lower.
Observation 1.

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Observation 2. Wss decreases as the PRM's ' predicted sigmoid score v.s, rises. Thus, Way has positive correlation with the PRM's prediction of a step's correctness.
Observation 3. UK + when St: reaches the correct answer.
Tek + 0, mi = 0, i.e. Uk converges to upper bound only
Based the features of Uk and ttsy we directly predict the quality value of partial solutions and guide search once we have a precise PRM and accurate prediction of my- In our approach, instead of separately training models predict rsa and mg. we simply train a process reward model V₂ to predict Ug2 serving as variant of common PRM. With reward incorporated in the calculation of uk. there is no need to separately train a reward model, saving considerable effort for answer selection.
Process Reward Model Guided Tree Search MCTS'. Tree search methods like [24] and [26] require value function and outcome reward model ra prune branches, evaluate final solutions and backup value. However, using ORM to evaluate final solutions and backpropagate means every search trace must be completely generated, which is costly and inefficient. Recent work [14] suggests using a learned LLM value function in MCTS so the backup process can happen in the intermediate step, without the need for complete generations. Their work greatly improves search efficiency but still relies on an ORM to select the final answer. Drawing inspiration from these works, we further propose a new variant of MCTS, namely MCTS", which uses quality value 2/. value target for trained LLM-based process reward model and guidance for MCTS as well.
Given the above properties, of any partial solution, select, and backpropagate in intermediate nodes. Aside from the use of quality value, also incorporate special Monte Carlo rollout method and self-critic mechanism to enhance efficiency and precision, which are explained detailedly in Appendix C.I. We express MCTS as an algorithm that comprises four main stages in each iteration, namely node selection, thought expansion, greedy MC rollout, and value backpropagation. Similar to common MCTS settings, the algorithm runs search tree T, for each single science reasoning question g. Every tree node C represents a series of thoughts or steps, where a partial solution pc, number of visits ne, and corresponding quality value ve are recorded. For simplicity, we denote each node tuple C (PC, nee Uc). An overall pseudo-code for MCTS* is presented in Algorithm 2.
directly use the process reward model Vo evaluate the quality
3.2 Self-Training Pipeline
As shown in Figure 1, based on the proposed tree search algorithm MCTS", we perform self- improvement on the reasoning policy and process reward model. After initialization of the policy ' and process reward model Vg. we iteratively employ them and utilize the search tree T, generated in the process to generate high-quality solutions for specific science or math questions and conduct a self-improvement process, called ReST-MCTS*. Our work draws inspiration from the MuZero [20] framework and applies it to the training of LLMs which we term "MuZero-style learning LLMs".
Instruction Generation. In this stage, initialization starts from an original dataset Do for the training process reward model Vg.
Collect process reward for process reward model. The extraction of new value data is relatively more complex, we derive the target quality value of partial solutions of every tree node near correct reasoning path on the pruned search tree T'. We first calculate my for every tree node C' that is on at least one correct reasoning trace (including the root) according to its minimum reasoning steps required to get to a correct answer in T Then, we use the hard estimation in Eq. (11) in [12] to calculate Tax i.e. Tsk - ,H: which means a reasoning step is considered correct if it can reach a correct answer in T' g. Using my and Tag, we are able to derive the value of the partial solution of every node on or near one correct reasoning trace. For each node C' (with partial solution PC = [81, $2, ... -8--1 1]) on at least one correct trace and a relevant forward step 8k. we can derive the value 24 using Eq. (1) and weighted reward Wi using Eq. (2), with m& set to the same as m HF. = 0 in Eq. (11). concrete and detailed example of this inferring process is shown in Figure We update all these rewards and values starting from the root and collect all (Q P. v) pairs to form Dy; in i-th iteration, which is used for training process reward model in the next iteration.
5

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Collect reasoning traces for policy model. As shown in Figure 4, the search process produces search tree T., consisting of multiple reasoning traces. We first prune all the unfinished branches (branches that do not reach a final answer). Then we verify other traces' final answers acquired in the tree search according to their correctness through simple string matching or LLM judging and select the correct solutions. These verified reasoning traces, as (A)=aaa") )[== (where N is the number of sampling solutions, A, the j-th solution, and a is the final correct answer) in i-th iteration, are then used for extracting new training data for policy self-improvement. This process is followed by Eq. (13) (i 2 1) to execute the policy self-training.
Mutual delf-training for Process Reward Model and Policy Model. Compared to previous work like ReSTEM EM [6]. which only concerns self-trate self-training for the policy and demonstrates that the policy can improve by iteratively generating new traces and learning from the high-reward ones generated by itself, work simultancously improves the process reward model and policy model self-training. With the process reward model's training set Dv, initialized and new problem set Da given, we can start the iterative self- self-training process upon Vi and We solutions for DG. with implement details illustrated in Section 3.1. In the i-th (i = 1, .... iteration, we train Vo with Dv,_, obtain Vi and train policy model RS on DG, generate new generator Ts, At the same time, De, drives the update of V; to Vi+1. We present iterative self- olff-training that the process reward model and policy model complement each other Algorithm 1.
perform MCTS and generate
Algorithm 1: Mutual self self-training ReST-MCTS' for value model and policy model.
Input: base LLM म, original dataset for policy model Ds. original dataset for value model Do new problem set De. number of solutions N, j-th solution Aj. correct solution a", value model
Ve. weighted value function w, quality value function v, number of iterations T.
1: T5. < SFT( Ds.) fine-tune generator
2: Dv, - generate_ valuv_vllldaaa sasee_dttaaaD Do, w, v) // initialize train set for value model
3: Vo < train_vall siti__aluu_mmoee(( Dv,) // initialize value model
4: for = to do
5: DG.
cannerate_policee data(Ts,,,, Vi-1 guided MCTS", DG. generate synthetic data for policy model
6: for j= N do
7:
Do.(A,=a')> label_ label_Lorrectness((()))) // match and select correct solutions end for
8:
9:
IS. + SFT(Ts, =g, // self-training policy model
10:
Dv, < extract extract_value data(DG, ) // collect process reward and extract value data
11:
V. train value model(V;-1, Dv,) self-training value model
12:
end for
Output: TSr VT
4 Experiments
We validate ReST-MCTS" from three perspectives:
Self-Training approaches which use generated samples and evaluated for multiple iterations, such as Re and Self-Rewarding. on in-distribution and out-of-distribution benchmarks under three LLM backbones, as shown in Table 2. ReST-MCTS" outperforms existing approaches in each iteration and continuously self-improves with the data it generates.
Process reward models which are compared with the state-of thettrt techniques, such MATH- SHEPHERD (MS) and SC + MS on GSM8K and MATH500, as shown Table 3. Results indicate that the ReST-MCTS* learns good PRM and our reward model implements higher accuracy.
Tree-Search policy which are compared on college-level scientific reasoning benchmark under three L.MSs, such as CoT and ToT, as shown in Table 4. We also evaluated under the same search budget on MATH and SciBench, such as and Best-of-N, as shown in Figure Results show the ReST-MCTS* significantly outperforms other baselines despite insufficient budget.

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Table 2: Primary results by training both policy and value model for multiple iterations. For each backbone, different selfn-taining approaches are conducted separately. This means each approach has its own generated train data and corresponding reward (value) model. Our evaluation is zero-shot only, the shw-shot baseline only serves as comparison.
Model
Self- Training Methods
MATH
GPQAnssrend
CEval-Hard
Ave.
Oth iteration (zero-shot)
20.76
27.27
26.32
24.78
Oth iteration (few-shot)
30.00
31.31
25.66
28.99
(Below are fine-tuned from model of previous iteration with self generated traces)
w/ ReST (1st iteration)
30.84
26.77
21.05
26.22
LLaMA-3-8B 3-8B-Instruct
w/ Self-Rewarding (Ist iteration)
30.34
26,26
25.66
27.42
w/ ReST-MCTS' (Ist iteration)
31.42
24.24
26.97
27.55
w/ ReSTEM (2nd iteration)
33.52
25.25
21.71
26.83
w/ Self- Rewardin (2nd iteration)
33.89
26.26
23.03
27.73
w/ ReST- (2nd iteration)
34.28
27.78
25.00
29.02
iteration (zero-shot) iteration (few-shot)
29.34 28.28
27.78
22.33
29.29
22.26
(Below are fine-tuned from model of previous iteration with self generated traces)
RcSTEM (1st iteration)
23.84
26.26
20.39
23.50
Mistral-7B: MctaMATH
W/ Self Self-Rewarding (Ist iteration)
25.70
27.78
19.74
24.40
w/ ReST-MCTS' (Ist iteration)
31.06
26.26
17.11
24.81
ReST (2nd iteration)
26.26
24.16
Self Rewardiw (2nd iteration)
23.90
26.77
25.00
25.22
ReST-MCTS* (2nd iteration)
24.40
28.79
26.32
26.50
Oth iteration
25.18
23.74
$1.97
33.63
(Below fine-tuned from model of previous iteration with self-generated traces)
RcSTEM (Ist itcration)
22.72
24.75
51.32
32.93
McGG
Self Self-Rewarding (Ist iteration)
22.50
26.26
47.37
32.04
ReST-MCTS- (Ist iteration)
24.86
25.25
51.32
33.81
w/ RM (2nd iteration)
25.86
25.25
48.68
33.27
w/ Self-Rewarding (2nd iteration)
23.86
28.79
48.03
33.56
w/ ReST-MCTS" (2nd iteration)
23.90
31.82
51.97
35.90
4.1 Initialization of Value Model
To obtain accurate feedback from the environment, we build the value model's initial train set Dy, from set of selected science math questions Do using process reward (value) inference, with human labeling process required. Then, we finctune the ChatGLM3-6B [27; 28] and Mistral-7B [29] model on this dataset, respectively, obtaining initial value models that, variants of PRM, guide the LLM tree search for higher-quality solutions upon both math and science questions.
Fine-grained dataset for science and math. Aiming to gather value train data for science, we integrate questions of a lean science dataset Daes within Scilnstruct [10] into Do- This dataset consists of 11 11,554 questions, where each question is paired with correct step-by-step solution. For each question (i 1,2, .N) and corresponding solution s(0) $1.2... K. Dacis we (t)
extract all partial solutions form samples d!!) (g))) $1.2.... .A (P) (0) = 1,2, ..).. To make the value model distinguish false steps, we also employ LLM policy (ChatGLM2) that (6)'
is basically incompetent for reasoning tasks of this difficulty to generate single steps k441 given
g()) and (i) obtaining new partial solutions Pk+1
(6)' [s . 9 and new samples d()
())
s s $1.2.... .1,,..... ky 6
241, 10 2,). For simplicity, the generated steps are regarded as incorrect. In
(2)
total, we collect 473.4k samples for training the initial value model. Afterward, we derive target
quality values for all samples d()) and de and use them to construct DVg- which is illustrated in Appendix B. We adopt an alternative method to generate value train data for math, as shown in Appendix B.I.

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RoST MCTIC
0.22
FRN Scrve-l
0.45
Accuracy
-..00
Accuracy
0.18
0.35
0.16
0.14
0.30
0.12
0
10,000
20,000
30,000
40,000
0
18,348 Completion Tokens (Average Per Question)
9,174
27,322
Completion Tokens (Average Per Question)
Comparison of value model on SeiBench. Figure 2: Accuracy of different scarches MATH and SeiBench with varied sampling budget.
Self-training value model MATH.
Table 3: Accuracy of different verifiers on GSM8K test set and MATH500. SC: Self- Seff-Consistency.
MS:
Verification is based on 256 outputs.
Models
Dataset
SC
ORM
SC+ORM
MS
SC MS
SC ReST-MCTS" (Value)
GSM8K
83.9
86.2
86.6
87.1
86.3
87.5
Mistral-BB: MetaMATH
MATH500
35.1
36.4
38.0
37.3
38.3
39.0
4.2 Evaluating Self-Improvement of ReST-MCTS'
In order to thoroughly examine the influence of ReST-MCTS' self-training on varied backbones, we execute 2 iterations of self-training and compare two representative self-training approaches, ReSTEM, which compares outcome reward with ground-truth answer, and Self-Rewarding, which judges outcome reward by LLMs, upon different base models, namely LLaMA-3-8B Instrucct [30]. Mistral-7B: MetaMATH [29; 31] and SciGLM-6B [10]. Primary results are shown Table 2. Concerning the dataset for sample generation, since we are primarily interested in the continuous improvement ability ReST-MCTS" in specific domain, we mainly include math questions in the dataset. For simplicity, we use the same dataset DG in each iteration. It involves questions selected from a train set of well-known benchmarks including MATH, GSMSK, and TheoremQA [32]. With the policy and value model trained simultaneously on samples generated from De, we observe that our self-training paradigm enables continuous enhancement the capabilities of both models on in-distribution and out-of -isttii-ittiotoo benchmarks, regardless of which backbone is used.
Iterative performance improvement on policy model. Previous LLM self-training approaches mostly rely on the generating responses of LLM and assume each question with the correct solution a high-quality sample while the intermediate reasoning steps are wrong or useless in many cases. Therefore, we compare the ReST-MCTS- with recent self-training paradigms by generating new samples under different reward (value) supervision strategies. For RM and Self-Rewarding, the default sampling strategy is generating CoT data, with generated data refined according to ground truth or reward provided by the policy, respectively. In comparison, ReST-MCTS- generates data samples via MCTS". with data refined referring to quality value and ground truth. The results in Table 2 show that all three backbones can be continuously self-improved by data generated by itself, using ReST-MCTS" as a paradigm. ReST-MCTS* significantly outperforms previous self-training methods EM and Self-Rewarding basically each iteration. This means the ReST-MCTS* can screen out self-generated data of higher quality for better self-improvement.
Iterative performance improvement on reward model. We also compare how our iterative trained policy and value model can improve the overall search results under the same token usage on the test set of MATH [33]. See implementation details in Appendix E.3. We show results in Figure 2 (a), where ReST-MCTS' (Iter #1) greatly outperforms most baselines but does not completely surpass Self-Consistency.
comparison, after more iterations of self-training, verification based on the enhanced value model basically outperforms Self-Consistency every point, achieving the highest accuracy of 48.5% that significantly exceeds the 42.5% of seff-Consistency. This indicates the effectiveness of our self-training pipeline.

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Table 4: Overall performance comparison with representative models SciBench.
Models
Subject
Chemistry
Physies
Math
All
Method
adkins
cherme
quan
matter
fund
class
thermo
diff
stat
cale
Ave.
CoT
23.07
8.82
2.12
7.46
10.00
28.57
12.68
GLM4
ToT
11.21
23.07
8.82
12.24
22.22
6.38
5.97
12.00
25.33
30.95
15.82
ReST-MCTS
13,08
28.20
14.70
8.16
22.22
4.25
7.46
12.00
26.66
30.95
16.77
CoT
5.60
7.69
5,88
6.12
6.94
2.12
2,98
4.00
16.00
11.90
6.92
GPT-3.5-tubo
ToT
8.41
12.82
11.76
6.12
11.11
0.00
0.00
10.00
18.56
9.52
8.44
ReST-MCTS
5.60
12.82
11.76
6.12
6.94
8.51
2,98
10.00
24.00
11,90
10.06
CoT
2,80
2,56
2.94
2.04
2.77
2.12
0.00
2,00
2.66
2.38
2.23
LLaMA2-13B Chat
ToT
0.93
5.12
2.94
4.08
2.77
0.00
1.49
0.00
4.00
2.38
2.37
ReST.MCTS'
5.12
2.94
4.16
0.00
4.00
2.38
2.90
4.3
Evaluating Reward Guidance and Reasoning Policy of ReST-MCTS'
Our main hypothesis in this paper is that better search policy getting higher-quality traces can improve self-training. In this section, we mainly focus on whether our process reward guided MCTS" can gain improvement to get better samples over different reasoning tasks. We first evaluate the effectiveness of the value model itself standalone in Table 3 and then evaluate the performance of different reasoning policies in Table 4.
Performance Comparison of Various Verification Models. As [1] suggested, different value models or reward models vary in accuracy and fineness. We perform tests on the questions of the GSM8K and MATH500 using multiple reward models and verification methods. It is worth noting that we include the same experiment settings of MATH-SHEPHEEE SHEPHERD (MS) [12] as a comparison since it also adopts an automatic train data generation method for reward models. For SC+ReST-MCTS", we utilize the same ToD-d sampling strategy MS, except that SC is performed according to own value model's 's output rather than the reward model of MS, which makes this direct comparison of different reward model training approaches. We record the model accuracy of Mistral-7B: MetaMATH on the selected test set, which is as shown in Table 3. Results indicate that compared to MS and SC+MS, SC+ReST-MCTS* (Value) exhibits higher improvement in solution accuracy on both GSM8K and MATH. This confirms the effectiveness of our value model, further indicating that our definitions of quality value and weighted reward are valid or possibly even better.
Performance Comparison under the Same Search Budget. Though the MCTTSbased search methods demonstrate significant improvement in model performance, they often require consid- erable amount of token input and completion, which makes it quite costly in some circumstances. Therefore, conduct more experiments to investigate the relationship between search token budget and model performance on science questions selected from SciBench comparing ReST-MCTS* and the same baselines employed for MATH, which are elaborated in Appendix E.3. Since our self-training procedure is primarily conducted on math data, so we do not consider the effects of self-training in this case. However, we point out that this can still be further investigated as study of transfer learning for self-training paradigms. Figure 2 (b) shows the accuracy of different approaches on SciBench when the completion budget changes. Results indicate that the ReST-MCTS' greatly outperforms other baselines despite insufficient budget. We notice that although CoT-based methods can improve greatly by increasing the sample budget, they tend to quickly converge to limited accuracy, which is not as satisfying as the ReST-MCTS*.
Performance Comparison of Different Reasoning Policies on Benchmarks. To evaluate the effectiveness of ReST-MCTS", we perform benchmark experiments on SciBench [34] in Tabel 4 and SciEval [35] in Table 8. All benchmark setups are illustrated in Appendix E.2. For the backbone of models, large-scale models GLM4 and GPT-3.5-turbo (both API), as well as a small-scale model are included. As shown in Table with the experiment repeated for times, we report the average accuracy scores (%) of 3 methods on 10 subjects. Concerning overall accuracy, the ReST-MCTS* outperforms other baselines for all 3 models, with GLM4 improved over 4.0% and GPT-3.5-turbo over 3.1%. On specific subjects such as chemme, quan, and stat, the ReST-MCTS* achieves significant improvement over 5.0%, indicating its great potential in discovering accurate solutions. Besides, we notice that our ToT baseline also performs well on many subjects, sometimes even surpassing ReST-MCTS*. This reflects that our value model can provide appropriate guidance for tree-search-base methods. We also discovered that for not very prominent. This reveals that small-scale policies may face difficulties when adopting complex tree search approaches since their capability for step-wise inference is relatively low.
1---A-111--Chh the improvement

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5 Related Work
5.1 Large Language Model Training
Large Language Models (LLMs) [36; 37; 38] have emerged as a notable success in various natural language tasks, Recent studies focus on improving the reasoning capabilities of LLMs, including collecting high-quality or larger domain-speciffc data [39; 40; 41; 42; 10; 43], designing elaborate prompting [22; 44; 45; 46], or training supervised learning [10; 31; 32; 47] reinforcement learning (RL) [48; 49; 50; 16]. When LLMs are trained with the RL algorithm. the generation of LLMs can be naturally expressed as the Markov Decision Process (MDP) and optimized for specific objectives. According to this formula, InstructGPT [51] has achieved remarkable success in optimizing LLMs to align human preferences by utilizing RL from Human Feedback (RLHF) [52]. RLAIF then uses AI feedback to extend RI from human feedback [53]. Our work aims to propose an LLM self-training method via process rewards guided tree search.
5.2 Large Language Model Reasoning
LLM reasoning algorithms include prompt-based chain-of-thought (CoT) [22]. planning-base represented by tree-of-thought (ToT) [24]. Scientific reasoning has several categories to mine the potential of existing large language models, resulting from different performances for problem- solving. Previous studies have attempted to outperform the direct generation. For example, in this paper [54], an approach for generating solutions in a step-by-step manner is proposed, another model or function is used to select the top-ranked answers, and ballucination is avoided by limiting the output to a narrower set. [55] presents a maicutic prompting inference method, which can generate abductive explanations of various hypotheses explained by recursion, eliminate contradicting candidates, and achieve logically consistent reasoning. -hain-of-thoughts (CoTT) [22] imitates the thought process like humans to provide step-by-step solutions given question. Self-Consistency CoT [23] improves the reliability and Self-Consistency of answers by sampling multiple interpretations from LM and then selecting the final answer that appears most frequently. Tree-of-Thoughts (ToT) [24] further generalizes the CoT methodology by considering multiple different reasoning paths in the tree and exploring coherent units of thought execute thoughtful decision-making. In our work, benchmark hard science reasoning tasks against [22; 24; 34; 35].
6 Conclusion
In this paper, we propose ReST-MCTS", self-training both policy and process reward model by high-quality samples generated by reward guided tree search. Inferred rewards from the previous iteration are able to refine the process reward model and self-train the policy model with high-quality traces. Experimental results show that the ReST-MCTS" outperforms other self-training paradigms and achieves higher accuracy than previous reasoning baselines under the same search budget.
Limitation: We discussed limitation in detail at Section H in Appendix. In summary, we need to show the ReST-MCTS' can generalize to other reasoning tasks outside of math (like coding, agent, ete); and tasks without ground-truth (dialogue, SWE-Bench [56]. etc). We also need scale up the proposed value model and further improve the data filtering techniques. One potential idea is to incorporate online RL algorithms that can help perform better self self-training for value models and policy models.
Acknowledgments
Dan and Sining would like to thank Zhipu AI for sponsoring the computation resources used in this work. Yisong is supported in part by NSF #1918655. Yuxiao and Jie are supported in part by the NSFC 62276148, NSFC for Distinguished Young Scholar 62425601, a research fund from Zhipu, New Cornerstone Science Foundation through the XPLORER PRIZE and Tsinghua University (Department of Computer Science and Technology) Siemens Ltd., China Joint Research Center for Industrial Intelligence and Internet of Things (JCIIOT). Corresponding authors: Yisong Yue, Yuxiao Dong, and Jie Tang.
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Appendix
Table of Contents
A Preliminaries
16
A.1 LLM Reasoning
16
A.2 Reward Verification
17
A.3 LLM Self- Self-Training
18
B Deduction Demonstration
19
B.1
Definition of Weighted Value and Quality Value
19
B.2 Detailed Deduction Weighted Value and Quality Value
20
Algorithm Detail and Process Example
21
C.I
Algorithm Details MCTS'
C.2 Data Generation Process and Specific Example for Reward Inference
24
Model Comparison
24
Experimental Details
24
E.1
Training and Evaluation of Initial Value Model
24
E.2
Benchmark Setup
25
E.3
Baselines of Search Verification
25
E.4
Value Model of ReST-MCTS' on SciBench
26
E.5
ReST-MCTS" on SciEval
26
F Prompt and Instruction Examples
27
G Further Preliminaries of MCTS and LIM Reasoning with MCTS
29
G.1 Monte Carlo Tree Search (MCTS)
29
G.2 LLM Reasoning with Monte Carlo Tree Search
30
Limitations
30
Broader Impact
Reproducibility
A
Preliminaries
In this section, we briefly describe LLM reasoning, reward verification, and LLM self-training. The definitions for notations are in Table and model comparison in Figure 6.
A.1 LLM Reasoning
The use of reasoning approaches can significantly improve LLM problem- abilities [10]. Given a policy model, (an autoregressive pre-trained language model) and an input problem Q, m can autoregressive generate an output sequence & = (81, $2, ... 8x) by predicting the next token. The conditional probability distribution of generating the complete output sequence is:
(s|
( Q).
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Table 5: Notation Table.
Character
Mcaning given question/problem decoded answer final correct answer
solution
partial solution
sk
k-th step of solution
K
number of reasoning steps of solution j-th solution
4,
N
number of solutions
d
number of preference pairs
Ta
common PRM reward of a single step 8x. used to define weighted reward
Way
weighted reward of single step #4: inferred in self-training process after trace generation quality value partial steps Pk. used to guide search; inferred in self-training process reasoning distance of partial steps pk
%
mg
base language model original training dataset process reward model outcome reward model
Ds
Vo
re
Any problem can be reasoned by zero-shot prompting, few-shot prompting [57], chain-of-thought
(CoT)) [22], self-consistency CoT [23] or best-of-N (BoN) selection [1], tree-of-thought (ToT) [24],
Monte Carlo tree search (MCTS) [14], graph-of sogh-offthhoght (GoT) [58], amongst other approaches. Generally, recent studies represented by CoT [22] aim to improve the overall performance follows:
Pa = a² Q) B(so....-...x)/// P.1I). P(A a" 80, $1, ... .8K. Q))
We often call each trajectory (81 82, $2, 8x) reasoning trace. P(A a" 50: 81 ................................ KK, Q) is the probability to get correct answer a" given a problem Q and a reasoning trace s. Given a original training dataset D {Q1, Q2. Qu ). dataset can be produced by sampling N times per problem Q using the above-mentioned reasoning strategies:
Ds. {(Q2, s1) (Orr 5M)
(5)
As shown in Table STaR [4], RFT [5], ReSTEM [6], V-STaR [7], and Self-Rewarding [16] adopt CoT prompting. Step-by-step [1] and MATH-SHEPHERI [12] leverage the best-of-N selection reasoning evaluation strategy. TS-LLM [14] utilizes MCTS as a reasoning policy to fully generate
traces. Our work similarly seeks correct reasoning path to maximize the expected cumulative P.
A.2 Reward Verification
In the context of LLMs, we assume that the reasoning trajectory derived from the policy model sampling. The common first step of self-training methods is fine-tuning the base model on the original dataset Ds, and obtaining a new generator TS. Besides, a reward r considered to evaluate the value of the history trace. The objective of reinforcement learning (RL) with reward r(Q,s) s) is:
CRL. Ds, (rareneeeir((()))))
Recent works [1; 12), through PRMs and ORMs, model the objective of reasoning as a search to find the highest cumulative reward trajectory
problem Q and infer the final answer A.
ORM (D x + R) is trained with a cross-entropy loss:
LORM = A, r3 + (1 A.) log(1 Tz);
where A, is the golden answer (A, 1 if is correct else A, 0) and r, is the ORM's output sigmoid score. STaR [4], RFT [5], and ReSTEM [6] consider the outcome reward value label A, compared with ground truth answer. In V-STaR [7], its outcome reward is generated by multi-iteration LLMs, and reward r is defined via verifier Ty and LLM generator TS, follows:
r(Q, s) = 3 log #y (s ())
(8)
(s(Q)
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3 is the hyper- parameeer that controls the proximity of the reference policy TS,- Different from V- STaR, the outcome rewards in self-rewarding [16] is generated through LLM-as-a-Judgee prompting using dataset Ds, and policy model S.. In both V-STaR and self-rewarding, collected DVER is built for training verifiers as follows:
DVER= = ((Q), (0: (7), (Qj, j.t: 25,0)3=11
d is the number of preference pairs. However, previous works [1; 12] suggested PRMs demonstrate better supervision than ORMs among false positive solutions and provide more reliable feedback.
PRM (D x S + R is trained with:
K
LPRM Σ As, log Fay + (1,As,, log(1 rsa))
(10)
where As, is the golden answer (As, = 1 if S5 is correct else As, = 0) of 8& and Ts, is the PRM's output sigmoid score. Specifically, [1] regards PRM training as three-class classification with costly human annotations. Similarly, MATH-SHEPHERD [12] collects random rollout trajectories via BoN reasoning policy and synthesizes process rewards to construct the PRM training dataset autonomously, MATH-SHEPHERD defines the automated quality Tag. the potential to deduce the correct answer, for each reasoning step 8% through hard estimation (HE) and soft estimation (SE), which are,
{
1, BAj E A',
HE 86
(11)
0,
Otherwise
I(A =a")
and
ASE 5x
(12)
A" - and N is the number of solutions. To more precisely find reasoning trajectory with the highest expected reward, RAP [59] utilizes MCTS to estimate the expected future reward via state-action function (Q S x A - F) in traditional RL for each node. However, the decoding process of incorporating MCTS into LLMs is costly, because estimating and updating a state-action function requires recursive visit, and reward values need to be calculated by LLMs. The concurrent work pDPO [13], though effective, does not take internal accuracy generated by LLMs into account and ignores the number of steps to be generated. In this paper, we integrate process reward guidance with tree search to explore efficient solution space and synthesize high-quality trajectories.
A.3 LLM Self- Tel--Trinnng
Generation. Given new training dataset DG. self-training methods use generator Tt generate reasoning steps s and final answer A per problem Q. In each iteration (i 2 1). STaR, RFT, and ReSTEM check the generated solutions Do, with the binary correctness label z and keeps the correct solutions (A, = a") )1p=11 as DG.(A,=a")) )1 Based on the continuous iteration on positive samples, V-STaR and Self-Rewarding keep the correct and incorrect generated solutions per problem Q and train preference data pairs on constructed verifier data DVER with all data DG so the कv can learn the error patterns produced by generator in each iteration i. Then, the generator ITS,- here is T is fine-tuned on new generated dataset (A, = and again is updated as generator тS, This process is continuously running in subsequent iterations. Their iterative process and reward value are as follows:
Ds. " TSP DG (G,(A, where for RFT and i i for STaR, ReSTEM V-STaR, and Self-Rewarding.
T3:-1
TS.
(13)
Improvement. The practical way to accomplish reasoning tasks on Ds, is supervised fine-tuning (SFT) that trains policy model by minimizing the negative log-likelihood loss on the training dataset:
T
LsFT (T) = -E(
(og Q)]
(14)
Recent offline preference learning methods replace LLM verifiers (before being trained on LLM generator and binary classification) with DPO [50]. The training DPO objective for a verifier my is described as follows:
Coro(Ty: Ts,) = an) (log ar((2,**)
r(Q.s s ))}
(15)
where a is the logistic function.
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B Deduction Demonstration
B.1 Definition of Weighted Value and Quality Value
Weighted Value. Recall the definition of the weighted reward:
Uk-1 2r k= 1,
Uss my +
(16)
And we know that Tak E [0, 1]. Now, let's examine the maximum possible value of the term (1 )r ). Since Tay E [0, 1], the maximum value of (1 2r occurs when Tax = 0. In this case,
(1 2r,.)) = Therefore, we can conclude that -1 S (1 2r S...
Next, let's consider the denominator, (mx + 1). Since my = K - k. and K 2 k, we have my 2
and my 1 2 1. Therefore, we conclude that (mz + 1) 2 1.
Combining these results, we can rewrite the weighted reward as follows:
uk-1 (1 2r S 11 1k-1l -11 2r S 1
Usk
(17)
m
Hence, we deduce that Wss < 1/- which indicates that the weighted reward is bounded by the absolute value of the difference between and the previous quality value.
Quality Value. Recall that the quality value Up is determined by incorporating the previous quality value 1 and the weighted reward W/s, of the current step.
marr + WSs 0)
(18)
Now, we can inductively reach the conclusion that 2k2 E [0, 1]. starting from the fact that Up = 0. Assuming that Up-1 E [0, 1], then
derive 822 [0, using the bound of W'sz.
mox((U-1 + Wag. 0) S - + |1 -- =
(19)
Therefore, based on the properties of the weighted reward and the definition of the quality value, can deduce that UK is indeed confined within the range [0, 1]. k 0,
Fine-grained Dataset for Math. We adopt an alternative method to generate value train data for math. For this method, we only demand correct final answer a, for each question q, which simpler satisfy. Specifically, we integrate the MATH [33] train set into Do. For each question ( and answer ()) we use Mistral-7B: MetaMATH [31] as policy to generate solution traces in a simple -hattt--iist-search (BFS) manner, obtaining search tree Ty(3) similar to the one of the self-training process. Subsequently, we verify the obtained answers of all leaf nodes of T according to a, The verified search trees are then used to derive data samples with target values for Dv.-
Construction of value model training set. Previous approaches like [1] that employ PRMs usually require human annotation to initialize a train set, which is quite costly. comparison, our value model's 's initial training set can be constructed at lower expense.
For math data, we deploy the same approach mentioned in section 3.2 to infer process rewards and quality values of partial solutions within the verified search tree ()(3) While for science data, this value-inferring process is slightly different. We still derive the target value of ( based on the definition in Eq. (1) and Eq. (2). Under the assumption that original solutions are reliable and concise, we can simply regard (() as the globally optimal reasoning path for g()) Therefore, we derive that:
(4) k 0. m K, k. w (t) and 2
(20)
K;
K,
Derivation. Please refer to the detailed derivation for Eq. (20) in Appendix B.2.
(8) In contrast, for generated false samples, we set Tset1 = L, m = K4 k (since still K; k correct reasoning steps required to reach final answer). Considering that Uk = R. we have:
(6)'
k.
(9) = K; - k K
K,
w 1++1
(21)
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(R) U4+1
max(0,
k-1 K₁ K₂ (K; k: + 1)
(22)
Derivation.
Please refer to the detailed derivation for Eq. (21) and Eq. (22) in Appendix B.2.
Collecting all samples and their corresponding derived quality values, we acquire the initial training set Dv, for value model Vg, described in Appendix E.I.
B.2 Detailed Deduction for Weighted Value and Quality Value
Here, we deduce the weighted reward using Eq. (2) and quality value using Eq. (1):
when k = 0:
0o 0
(23)
when lt
1-0
w1 K-1+1 (1-21 2
(24)
1
81 =
0)
(25)
K"
when k: = 2:
1
K - 1
U22 K 2
K
(1 - 2
K(K 1) K
(26)
1 + 0) = 2
2/2 =
(27)
K K
K
therefore,
1
1
Wk
Wk-1
Wk+1 K
(28)
K
K'
k
k -
+ 1 K
Uk
Uk
K Uk41
(29)
K
my = K k, m.
= K k + 1, m = K - 1: - 1.
(30)
Then, deduce the Eq. (21) and Eq. (22):
1 2k
Wh+1
(1
2r sht1 2r
m&+1 + 1
1 k
K
(1 - 2 1)
(K k) +1
K k
K(K - - 1)
1 K k
(31)
K K k+1
20

---

2k+1 = max 1kk + wk+1, 0)
1
max
K K - k: + 1 1
K
k(K - k + 1) (K - kk)
maxx
K(K - k +
k(K - +
1) 11
(K - - 1)
-max
0)
K(K - k 1)
(K l + () 1) 11+1
.max
.0)
K(K - ++
(K - k + +)(k -)
max
K(K - -k+1)
K(K k: + 1)"
max
K(K - 1 1++) 0)
(32)
Algorithm Detail and Process Example
C.1 Algorithm Details of MCTS*
Node Selection. Similar to [14], we propose to start each selection process from the initial root, since this allows backtracking. Within each iteration, the node selection stage is first executed, where leaf node Csetect is hierarchically selected starting from the initial root. To incorporate the quality value of nodes, we use UCB as the criterion to select child rather than the UCT [26], which is as follows:
Aparent
UCB(C) = cc + +eV
(33)
TCC
where Rpporntt is the number of visits of the parent node of C, E is a exploration constant. For each intermediate node, we select its child with maximum UCB. This criterion considers both quality value and visit count, thus it encourages the exploration high-quality nodes while leaving some opportunity for underexplored nodes.
Thought Expansion. Secondly, the value of the selected node Csntect is compared with a threshold I (in our experiments, / it is set to 0.9). If the UCCetees >= 1, the node's 's recorded partial solution .Ceeeee [81, 82, , is deemed acceptable as final solution (since ti get close to only when C close to correct final answer), which is then directly returned output, terminating the algorithm. This is different from the method adopted by [60] since no reward model estimation is required. Otherwise, the expansion stage is initiated, where new solution steps 1, = 1,2, .... b) are sampled by prompting the policy TS. i.c. 8,41, ~ T (,.,,,... ,A (4) b is the number of samples or branches. Subsequently, new nodes C, ([$1, $2, , Sk, Sk+1,s), s. 0, vc.) are added to T2 with uc, assigned by the value model, VC. + Ve (q). Note that we also incorporate a self-critic mechanism into this expansion process, which will be illustrated later.
Greedy MC Rollout. [60] and [14] use a simplified three-stage iteration that doesn't include a simulation process on leaf nodes. In contrast, we believe that simulation process still brings about useful information for value estimation, despite the rise in generation and time cost. In this stage, we propose to simulate few steps upon the new node C; with maximum predicted value. Reasoning steps starting from this node will be sampled step-by-step and evaluated, while only the most valuable path is further explored, until step limit m is reached. The highest quality value acquired in the sampling process Umax is recorded and used to update Uc, with a weight parameter a following:
vc, + avc. + (1 - a) ammmar- Besides, the visit count ne, is also updated by nc, + nc, +1.
(34)
Value Backpropagation. Finally, we conduct value backup starting from Csetect- The value of every parent node of Ceetect is updated using a weighted average method. For every node C on the trace from root to Csclect, we update its ne and ve as follows:
TC nc + 1
(35)
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Algorithm 2: The proposed value guided search algorithm MCTS".
Input: question 7, inference model &. malue_modes Vo, max iteraattsss T, threshold 1, branch b, palloutssteps m, roll branch d, weight_parameter o.. Tg Initiialize tree(q)
2: T.. Ve initialize_modelllitii
Ve)
3: for in range(T") do
4: C' r
5:
Node Selection
6:
while Ct is not leaf node do
7:
C + argmazc' Echillren(C)) + +EV "co"
// Select child node based on UCB In
8:
end while
9:
if vc 2 then
10: Return pC // Output solution
end if
12:
Self-Critic
Get as. response for further exploitation or o DD _Do_self____ critic(pcllg Ig, TT.)
stop reasoning
if o is not Eol then
15:
Thouugh Expansion
for j in range(b) do
17:
Cj < Geeeeeeeeewwa_eeeeeeeeeee _e___eiiw_cdd_,_.oe pc Ts,) // Expand based on previous steps and self-critic
18:
VC, + Vo(pc,la) // Evaluate with value model end for
19:
20:
Greedy MC Rollout
21:
C' argmaz EC)i (C) (v
22:
p=Pc
23:
Umaz =
24:
for k in range(m) do
25:
p, Umar k- and record the best-observed value end for
step_with____t wish_best_______ valuelalullnee s.. Vo, // Sample children
26:
27:
VCc' + aucc + +1 Ummma
28:
The &nd +1// Update value and visit count of the rollout node
29: end if
Value Backpropagation
31: Back propagate(C)) // Update value of parent nodes using weighted average
end for
33:
C=Get_bestL best_node(T,)) // Fetch the node with the highest value in the search tree
34:
Return PC
Output: PC
and
WC < Emi, UC, Emc.
(36)
where Ciii= ..,,,...,)) are the children C. This actually updates the value of C' according to its children's value expectation.
Determine Termination via Self-Critic. Although the value model provides accurate evaluation for partial solutions, it cannot consistently signal logical termination, especially when the inference model reaches false conclusion. Consequently, even when false final answer is generated, further exploration beneath this node may still be conducted, leading to reduced search efficiency. Therefore, we propose to use self-critic to provide extra timely signals logical termination that avoid unwise prompt the inference model to
exploration well as insight into deeper search. Specifically, generate End of Inference (Fol) signal or offer advice o on following exploration steps based on existing partial solutions p before each expansion stage. The expansion and MC rollout stage will be
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Q: Suppose that roil two fair 6-sided dice. What is the probuhility that the two numbers rolled sum to 4?
- 0.25
-0.25
0.33
Step Each die six sides, there six possibilities for the outcome of euch die, Therefore, the number of outcomess the two diee is 6*555-5 5-30.
Step I: First, we need to determine the possible outeomes of two dice. Each die has sides, so there are six possibilities for the outcome of each die. Therefore, the total number outcomes the two dice is 6*6m36
Calculate the probability. The probability that the sum of the two dice will be is 6/36-1/6.
Step I:
m
m 3
U
-0.25
0.33
w 0.22
Step Suppose sum of the two diee is We list all the resalts that satisfy the condition: (1,3),
Step 2: Determine the number of results that satisfy the problem condition. According the sum of the two dice 4. We can list all the results that satisfy the condition: (1,3), (2.2), (3, number results that satisfy problem condition is 3.
Step 2: Next, we can list all results that satisfy the condition: (1.3). (2.2). (3.1). So the number of results that satisfy the problem condition is
problem.
(2,2), (3,1), (4,0),
number of results that satisfy the peoblem
condition is 4, Thus, the probability is 4/30-2/15.
m = 2
V 0.56
0.67
W 0,22
W 0.33
0.17
Step 3: Calculate the probubility. The probability that the sum of the two dice will be is 4/36-1/9.
Step 3: Calculate the probability. Based on the preceding steps, we already know that the number of outcomes satisfying the problem condition is 3, and the lotal number of cutcomes of the two diee is 36.
Step 3: The calculated
probability can be obtained by dividing the number outcomes satisfying the
condition by the number of rotal outcomes.
== 0.50
Therefore, the probubility is equal divided by 36, which is 1:12.
0.78
Therefore, the probability that the
sum of the two diee will be is 1/12.
W 0.33
Step 4: Therefore, the probability that the sum of the two diee will be is 3/36=1/12.
m
Figure 3: Detailed inferred process of a concrete example. The search tree Ty has already been pruned during this stage, with traces verified. Starting with the inference of m. we gradually update all weighted rewards w for actions (steps) and quality value v for states (partial solutions). Taking the false Step
an example, since makes a mistake in calculation, it still requires the same number of steps as its parent to reach the correct answer (one step to correct the calculation mistake), i.e. m 1. As no trace starting from this node reaches the correct answer, we have r. 1. Thus, we derive w 1-0.67 (1 2 1) = -0.17, and therefore v = max(0.67 - (-0.17), 0) = 0.50.
1+1
skipped if an Eol signal is received. Otherwise, the advice will be utilized in the following expansion stage as part of the inference prompt, so vS, generates new steps based on both o and p. An overall pseudo-code for ReST-MCTS' is presented in Algorithm 2.
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Search
Prune
Verify
Infer reward&value
END: Inference
weighted vi quality valuo
Figure 4: Detailed process of new sample data generation for the self-training framework.
C.2 Data Generation Process and Specitic Example for Reward Inference
The data generation process of our self-training approach consists of mainly four stages, namely search, prune, verify, and reward inference, which is demonstrated in Figure 4. For reward inference, detailed example shown in Figure
D Model Comparison
ReST-MCTS" vs. AlphaLLM. As an approach that aims to enhance LLM inference, AlPhaLLM [61] utilizes a tailored MCTS algorithm and critic models to provide precise feedback. Even though AlphaLLM also adopts MCTS and critic models for self-improvement, their approach is different from ours in various crucial aspects, as elaborated below.
(1) Design of MCTS algorithm. For the level of search, AlphaLLM's 1yMCTS considers options as action, with termination signals delivered by a termination function 8. contrast, we use reasoning steps as action, which is achieved through tailored prompt design. Concerning critic models, we use single value model provide evaluation for intermediate nodes. The model is trained to predict specially designed quality values that reflect completeness and correctness of partial solutions, rather than estimating the conventional definition of value function in RL. In addition, we also incorporate self-critic mechanisms into the tree search algorithm to provide insights for the policy (Appendix C.I), which AlphaLLM does not adopt.
(2) Definition of reward/value. Our definition of weighted reward and quality value is novel, leading to significant differences between our method and AlphaLLM across various processes such as critic model training, data synthesizing, and data filtration. Since our design of quality value involves information on process reward and reasoning distance, our value model trained on this target can naturally provide sufficient feedback during the search, with no need for implementing other critic models mentioned by AlphaLLM.
(3) Self- Seff-Trainng algorithm. Although AlphaLLM also includes iterative self-training, the imple- mentation method varies greatly. Most importantly, their critical model is static throughout the iterations, which means they focus more on the improvement of policy. In comparison, we also consider the impacts of self-training on the critic value model. As demonstrated in Algorithm we calculate process rewards and quality values according to the final search tree of questions within each iteration, which are then used as new training data for the value model.
E Experimental Details
E.1 Training and Evaluation of Initial Value Model
Initialization of Value Model. We split Dy, and use the train set to finetune ChatGL and Mistral-7B to predict the value of partial solutions. We simply add linear layer to the model to
24

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directly transform probabilities to scalar value. Moreover, we use the AdamW optimizer [62] and MSE loss Eq. (37) to optimize, eventually obtaining an initial value model Vi that can evaluate the correctness and completeness of step-by-step solutions. Note that the learning rate is set to le-6 in this process. The MSE training loss is shown below:
LMSE E(a (Vo - 0/2.
(37)
Evaluation of Value Model. We use the test set containing 14k data samples to evaluate the value model with an absolute tolerance of 0.1:
Accuracy = - Mi= [((clip(Ve(xvla))
-0l < 0.1)
(38)
where & is the number of test data samples, qi is the question of sample pi is the partial solution of sample i and vg is the target value of sample i. Our initial value model achieves an accuracy of 69.3%, which means it is reliable in most situations. We also conducted study measure the value model's performance on science benchmark SciBench [34] compared to outcome-supervisee reward models and self-critic methods in Table 7.
E.2 Benchmark Setup
To compare the performance of different search methods, we construct a standardized benchmark test that can be generally used on labeled science or math datasets like MATH, SeiBench, and SciEval. Aside from the ReST-MCTS", we incorporate two other baselines: Chain-of-thougle (CoT) and tree-of-thought (ToT)). For each method, specialized prompts P are designed to execute the search process. Besides, inference model and value model V are deployed to provide deduction and feedback. Concerning the CoT baseline, we use Celf-Consistency to calculate accuracy. For the ToT baseline, we use simple greedy depth-first-seane value model. The algorithm stops exploitation when a max depth of 10 is reached and ends when a node value exceeds the threshold 0.9. For ReST-MCTS' self-critic is used and the ending threshold also set to 0.9. The rollout step limit m is set to 2, a is set to 0.5, and the number of iterations T is set to 50 by default. Moreover, both tree search algorithms use b 3 by default, where b is the number of samples generated in the expansion process as mentioned in the former sections. After the search process, the policy is prompted to extract the final answer based the obtained solution, which is then compared with the ground truth to determine correctness. The results of these methods on benchmarks are illustrated in Section 4.3.
(DFS) algorithm with node values assigned by the
E.3 Baselines of Search Verification
The basic settings of relevant verification baselines are illustrated as follows:
ORM + Best-of-N (BoN) For simplicity, employ the ORM used by [10], which is trained on Scilnstruct. For each question, we sample solutions and select the solution with the highest ORM score output. N is used to control token usage.
ReST-MCTS* Implementation of ReST-MCTS*, using the value model Vo as PRM guide MCTS" The variable controlling token usage is the iteration number T and branch parameter b.
Self- Coff-Coniistecyy N solutions are generated for each question using simple CoT prompt. Their final answers are then extracted and classified, with the most frequently occurring answer selected as the final output. N is used to control token usage.
PRM Best-of-N (BoN) With value model Ve used as PRM. perform DFS-based tree search. Every selected solution s is evaluated by PRM score FPRM = π 1=1 U;- The one with the highest PRM score among all N solutions is regarded as the final output. Under this setting, b is set to while is used to control token usage.
As shown in Figure 2, we have compared the number of tokens consumed by each algorithm to achieve certain accuracy on MATH and SciBench. Results reveal that to reach a certain expected accuracy, MCTS* basically requires fewer tokens than other algorithms like SC and ORM + BoN. This means that based on the same expected standards, MCTS* outperforms other algorithms while maintaining reasonable computation cost. As for the actual running time, have recorded
25

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Table The average running time of different algorithms (under our basic experiment settings) on single question.
Method
CoT + SC
ORM + BoN
PRM + BoN
MCTS
Running time (8)
41
43
73
108
MATH
37.0
33.5
34.5
41.5
Table 7: Accuracy of different reward/value models on the questions selected from SciBench, they all use MCTS" as search policy.
Dataset
Models
ORM
PRM
Self Relf-Rewarding
ReST-MCTS' (Value)
GLM4
20.2
22.0
20.2
22.9
SciBench
GPT-3.5-turb
12.8
17.4
13.7
20.2
0.22
0.20
Accuracy
0.18
0.16
0.14
0.12
25,000
75,000
100,000
Total Tokens (Average Per Question)
Figure 5: Accuracy of different methods SciBench with varied total token usage per question. Both the completion token and prompt token are included.
the average running time of different algorithms (under our basic experiment settings) on a single question, as shown in Table 6. We see that MCTS spends more time exploration and simulation than other simple algorithms. However, since our method adopts different design of value, it does not require massive Monte Carlo estimations. This reduces the running time of our algorithm and limits the time consumption to a reasonable range. Notice that MCTS* can achieve high accuracy that other algorithms can never attain even at unlimited cost, we believe this extra time is fairly acceptable.
E.4 Value Model of ReST-MCTS" on SciBench
We employ the reward model obtained by [10] (which is used as a classifier for SciGLM) as the ORM and our fine-grained value model as PRM to provide the outcome reward and step-wise value respectively. We also include the Self-Rewarding method, where the policy model itself is instructed to provide step-wise value. For all methods, the number of samples for each step set to Using this setting, record the model accuracy of GLM4 and GPT-3.5 which are as shown Table 7. Results indicate that compared to ORM and Self-Rewarding, PRM- based methods exhibit higher accuracy. This confirms the effectiveness of our value model. In addition, Figure concerns the total consumption of the token budget, including all prompt tokens and completion tokens. However, still have to note that the total token usage (especially prompt tokens) of the ReST-MCTS* increases rapidly as
on the selected questions,
b and T risc.
E.5 ReST-MCTS' on SciEval
SciEval. Similar to SciBench, we perform benchmark tests SciEval. Results are shown in Table For both GLM4 and GPT-3.5-turbo, ReST-MCTS" again outperforms other baselines overall accuracy, with an accuracy of 79.87% and 62.31% respectively. However, we notice that though tree-search-basee methods demonstrate an advantage on average, they fail to improve the performance of the CoT baseline on some parts of SciEval. We examine the data distribution and discover that
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Table 8: Overall performance comparison with representative models SciEval.
Models
Method
Part
Part II
Part III
Part IV
Ave.
CoT
52.50
85.83
80.83
78.11
74.32
GLM4
ToT
60.83
87.77
84.16
79.77
78.13
ReST-MCTS*
69.16
88.05
83.33
78.94
79.87
CoT
28.88
78.61
70.00
71,46
62.24
GPT-3.5-Turbo
ToT 32.50
76.
68.05
67.59
61.06
ReST-MCTS'
29.72
78.88
69.72
70.91
62.31
these parts basically all single-choice questions. they are less difficult compared other types of questions, the Self-Consistency CoT approach may already be competent. Besides, these questions often require few reasoning steps, which may the main reason why tree search methods do not perform as well as expected.
F Prompt and Instruction Examples
We present some instruction examples used ReST-MCTS' and self-training process this section, including:
Inference instruction This instruction used in tree search for the policy to generate new steps based on previous self-critic information.
Self-critic instruction Used for generating the Eol signal or advice for further search.
LLM verify instruction This instruction is employed in the data generation process of self-training when an answer needs verification by LLM (GPT-4 for our case).
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RFT'STaR/ReST
CoT
Vi
select
bi
SFT
(x
M
M
V-STaR
CoT
select
by:",
SFT
M
(x,)
M
y!
DPO
yi
BoN
HE/SE
MATH-SHEPHERD
Vi+ , yi
PRM
select
Di
PPO
{x}
M
M++1
ORM
y()
CoT
Self-Rewarding
D
select
(x y y("
DPO
(x,}}
M
M
M
MCTS
Value Model
ReST-MCTS'
Vi
select
by"",
SFT
(x,]
M
PRM
M+1
y(")
Figure 6: Comparison between existing self-training methods with our proposed ReST-MCTS*.
This is an example of inference instruction.
Instruction: Given science or math problem, corresponding solution that may be incomplete as well as some advice on solving this problem, your task is to generate new solution step based on the advice and existing steps.
Problem:
Let and 6 satisfy ab 7 and b = 5. What is the value of a² + 6??.
Solution:
know that (a+ + b)2 25. Step 2: Thus, a² + 2ab + = 25.
Step I: From a b 5,
Advice:
From step we know the value of a² 2ab + b2. Since also know the value of ab, we can derive a² + b².
Policy Output: Since a² 2 25 and ab = 7, we derive a2 (a² (2) 2ab 25
11.
This is an example of self-critic instruction.
Instruction: Given a science or math problem and corresponding solution that may be incomplete, your task is judge whether the solution has already reached a final answer conclusion for the problem. the solution has already reached final answer or conclusion, you should directly output "Final answer reached". Otherwise, you should give some advice on how to solve the problem based on current steps or what to consider next.
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Problem:
Let a and b satisfy ab 7 and a b What the value of a² 62?.
Solution:
Step 1: From a + b = 5, we can know that (a + 6)2 = 25.
Step 2: Thus, a² 2ab + b² 25.
Policy Output: The solution has not reached a final answer. Advice: From step we know the value of a² 2ab + b². Since we also know the value of ab, we can derive a² b².
This is an example of LLM verify instruction.
Instruction: Given science math problem, corresponding step-by-step solution, and the true answer of the problem. your task is to verify the answer obtained in the solution with the real answer. If the answer obtained in the solution is equivalent to the real one, output "I', otherwise output '0'.
Problem:
Let and b satisfy ab 7 and b = 5. What is the value of a² + 6??.
Solution:
Step 1: From a + b 5,
know that (a+ + 6) 25. Thus, a² + 2ab + = 25.
Step 2:
Step 3: Since ab
derive that a² 1/2 (a² 2ab + b2) 2abb 25 14 11.
So the answer is 11.
Real Answer: The value of a² + 6² is 11.
LLM Output:
G Further Preliminaries of MCTS and LLM Reasoning with MCTS
G.1 Monte Carlo Tree Search (MCTS)
MCTS [63] is search algorithm for optimal fecision-making large and complex combinatorial spaces. This algorithm represents search spaces as search trees and works on the principle of the best-first search based the evaluations of stochastic simulations. This technique has been widely employed in multiple gaming scenarios and achieved tremendous success, such as AlphaGo and AlphaZero [64] for computer Go Game. The basic MCTS algorithm involves iteratively search process with four steps for building search tree:
(1) Selection. The agent, starting from empty tree's root node, traverses the search tree's visited nodes and selects the next node according to the given selection strategy until the scalable node or leaf node is reached.
(2) Expansion. If this algorithm arrives at an expandable node, expands the search tree by selecting an unvisited child node.
(3) Simulation. After finishing the expansion, if the current node is non- non-terminal state, the algorithm will conduct one or multiple independent simulations from the current node until it reaches the terminal state. In this process, the actions are chosen at random.
(4) Backpropagation. The node statistics on the path from the current node to the root are updated based on the search results. Note that the scores assessed based on the termination state achieved.
To trade off the less tested paths with the best strategy identified so far, MCTS maintains a proper balance between exploration and exploitation by maximizing the Upper Confidence Bounds for Trees
(UCT) when child node k is selected as follows, UCT = X + 2Cpy / /2lmm; where the first term,
XR. is the average reward form arm A and this term encourages the exploitation of higher-reward choices. It is generally understood that X5 to be within [0. 1]. In the second exploration term, Cp > 0 constant to satisfy the Hoeffding inequality with rewards the range of [0, 1] [26]. n is the number of times the current node has been visited and ny is the number of times child &o has been
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visited. Generally, nA = 0 produces UCT value of oo, so that all children of a node have non-zero probability and are considered.
G.2 LLM Reasoning with Monte Carlo Tree Search
L.Mss have been invented, used in the past for autoregressive text generation, and are now very great at reasoning. Reasoning algorithms include prompt-based thain-of-thought (CoT) [22], planning- based represented by tree-of-thought (ToT) [24]. which successfully achieved the LLMSs' reasoning performance improvement. ToT combines the power of tree search (e.g., depth/breadth--irs, search) as an algorithm and LLMs' power as heuristic to tradeoff evaluation and generation. Reasoning via Planning (RAP) [59] with Monto Carlo Tree Search (MCTS) performs reasoning exploration and obtains reward reasoning paths.
Recent studies [60] present that Monte Carlo Tree Search (MCTS) agents benefit from task tis-sppccffic extension and expansion of the research tree. Specifically, the MCTS agents provide appropriate selection strategies for the state of the visit to guide the upcoming search based on the evaluation results (e.g., rewards and number of times the node has been visited) produced by the rollout and backpropagation process. Its mechanism coordinates exploration and thought exploitation within search space, which is superior to traditional depth-first search (DFS) or breadth-first search (BFS) algorithms based on the Tree of Thought (ToT). Building on MCTS, some studies have also explored the ability of the reasoning agent to provide search guidance. In catalyst design, [65] proposed Monte Carlo Thought Search, using LLM for complex scientific reasoning queries. [59] presents Reasoning via Planning (RAP). which adopts MCTS as a planning algorithm and repurposes the LLM as both a world model and a reasoning agent. Others like [66] experiment using the value function, a byproduct of the Proximal Policy Optimization (PPO) process, to guide the token-level decoding based on MCTS. In general, these approaches improve LLM's reasoning ability, whereas their performance on some challenging science tasks remains unsatisfying. addition, this series of methods differs from our contribution, where we propose a value model approach as reward functions for optimizing the reasoning path and improving model output.
H Limitations
In this section, we discuss some limitations of the ReST-MCTS*.
Generalization to other tasks, especially those without labels. Similar to many existing self- training works, ReST-MCTS" also relies on ground-truth oracle labels in a supervised dataset to filter the responses in the first place; in the future, we need to show ReST-MCTS* can generalize to other reasoning tasks outside of math (like coding, agent, conversation, etc); in addition, for those very complicated tasks that require multistep plamning and reasoning (like implementing the whole software like SWE-Agent), which does not have ground-trute answers, we need to propose better way to collect reward feedback (from few human labeling and symbolic execution or solver), and train generalizable reward model that can work and help for wider range of tasks.
Scale and diversity of proposed value model. Although we trained a value model based on Mistral- 7B: MetaMATH that performs better than the most advanced value model MATH-SHEPHERD, larger scale value model backbone is still needed for better PRM training. In addition, the initial training set of the training proposal PRM was generated by SciGLM, model that focuses on mathematical and scientific reasoning tasks but still lacks generality. While the current PRM achieves the best results multiple mathematical and scientific reasoning tasks, such as MATH and SciBench, it's worth exploring more diverse training sets to expand into various fields in the future, such as code generation and agent planning.
Self-training data filtering techniques. As we mentioned in Section I, the quality of reasoning trajectory affects the effectiveness of self-training, and generating a high-quality training set plays an important role. Therefore, we train the iterative process reward model to guide the tree search direction to obtain high-quality trajectories. On the other hand, since well-trained value models can help filter out the top-k generated trajectories with the highest process values, we also expect that a stronger and larger LLM model as the backbone of the value model might help to gain more.
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I
Broader Impact
ReST-MCTS" aims to introduce general self-training approach that uses MCTS to automatically label and generate process rewards, which will help generate high-quality datasets and improve the reasoning capabilities of LLMs. Fine-tuning a variety of LLMs on synthesized high-quality datasets can directly improve the performance of value models and generators and help to avoid the cost of manually generating process rewards during the training process reward model. The disadvantage that single reward model cannot be scaled to multiple domains, and we can solve this problem by training various reward models together on various reasoning domains. We believe that on the whole, the advantages outweigh the disadvantages.
J Reproducibility
We have made significant efforts to ensure the reproducibility our all experimental results. The training code, tree search algorithm, and the evaluation details for the ReST-MCTS" public in repository.
Training. Detailed training information about the value model, self-training backbones, and experi- mental settings can found in Section 4.1.
Tree Search Algorithm. Regarding the enhanced tree search algorithm MCTS', please refer to the Algorithm and public code.
Evaluation. We organized all evaluations, including the iterative self-training and value model, variety of value models, performance comparison under the same search budget, and different reasoning policies. All details can be found Section 4.2 for self-improvement evaluation and Section 4.3 for value models and reasoning policies comparison.
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Reviews:
[{"review_id": "Qc7dPtuVOn", "reviewer": "Reviewer_7P64", "summary": "The paper introduces ReST-MCTS*, a novel framework for self-training LLMs using MCTS combined with process reward guidance. The core innovation is in addressing the limitations of traditional self-training methods, which often include incorrect intermediate reasoning steps despite producing correct final answers. ReST-MCTS* leverages a modified MCTS algorithm, that integrates a process reward model to estimate the probability that each intermediate step contributes to the correct final answer. This allows for the automatic generation of high-quality reasoning traces without requiring dense human annotations. The inferred rewards serve as value targets for refining the process reward model and selecting high-quality traces for self-training the policy model. Experimental results on benchmarks like SciBench and MATH show that ReST-MCTS* not only outperforms previous self-training approaches (e.g., ReSTEM, Self-Rewarding LM) but also enhances the LLMs' accuracy through iterative self-improvement.", "questions": "- Other than the experiment in Figure 2 showing that multi-iteration ReST-MCTS* significantly outperforms SC, the other experiments only demonstrate marginal gains over SC. What do you think is the reason behind this?", "limitations": "NA (Good Enough)", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Originality: \n  - Proposes an innovative integration of MCTS with process reward guidance.\n- Quality: \n  - Theoretical foundations are robust, and the methodology is supported by extensive experimental validation.\n  - Demonstrates substantial improvements in performance on multiple benchmarks, showcasing the effectiveness of the proposed approach.\n- Significance: \n  - The approach addresses a critical challenge in multi-iteration LLM self-training, enhancing the quality of generated training data.\nope", "weaknesses": "- Related Work: While this paper mentions related work, such as Xidong, et al.'s [1], it does not adequately discuss the differences between its handcrafted value target and TD(λ) in [1]. A more thorough comparison with above approach would provide better context and highlight the novel contributions of this work. (e.g. TS-LLM is missing in figure 6)\n- Clarity: The paper suffers from clarity issues, as many definitions and notations are introduced later in the text rather than upfront. Additionally, several important figures are placed in the appendix, making it difficult for readers to follow the key points without constantly referring to supplementary materials.\n\n[1] Feng, X., Wan, Z., Wen, M., Wen, Y., Zhang, W., & Wang, J. (2023). Alphazero-like tree-search can guide large language model decoding and training. arXiv preprint arXiv:2309.17179.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks a lot for acknowledging the strengths of this work as an innovative self-training method, robust theoretical foundations, and extensive benchmarks. We have given a detailed discussion with related work, clarity issues, and the performance of SC.\n\n```\nW1: Concerns on lack of comparison between proposed ReST-MCTS* and TS-LLM.\n```\nIn Table 1, this paper initially contrasts TS-LLM with our proposed ReST-MCTS$^*$ concerning reasoning policy and reward guidance. To offer more depth, we have conducted a comprehensive comparison between these two approaches, aiming to offer a richer context and underscore the unique contributions of our work.\n\nAs an approach that aims to enhance LLM inference decoding and training, TS-LLM utilizes a tailored MCTS algorithm and value function to provide precise feedback. Their work proved that TS-LLM is indeed an efficient framework that significantly boosts the performance of LLMs without requiring extra data annotations. Even though TS-LLM also adopts MCTS and value function for self-improvement, their approach differentiates from ours in various crucial aspects elaborated as follows.\n\n**(1) Design of MCTS algorithm.**\nFor the level of search, TS-LLM considers each sentence or each token as an action, with termination signals delivered by a termination function $\\alpha$. In contrast, we use reasoning steps as action, which is achieved through tailored prompt design. Our proposed value model is trained to predict specially designed quality values that reflect completeness and correctness of partial solutions, rather than estimating the conventional definition of value function in RL. Additionally, we also incorporate self-critic mechanisms into the tree search algorithm to provide insights for the policy, as mentioned in Appendix C.1, which TS-LLM does not adopt.\n\n**(2) Definition of reward/value.**\nOur definition of weighted reward and quality value is novel, leading to significant differences between our method and TS-LLM across various processes such as critic model training, data synthesizing, and data filtration. Since our design of quality value involves information of $r_{s_k}$, $m_k$, and $v_{k-1}$, our value model trained on this target can naturally provide sufficient feedback during the search, with no need for implementing other critic models mentioned by TS-LLM.\n\n**(3) Self-Training algorithm.**\nAlthough TS-LLM also includes iterative self-training, the implementation method varies greatly. Most importantly, their reward model is ORM throughout the iterations, which means they focus more on the improvement of policy. In comparison, we consider the impacts of self-training on the critic value model. As demonstrated in Algorithm 1 and Appendix C.2, by carefully designing the data synthetic process, we calculate process rewards and quality values according to the final search tree of questions within each iteration, which are then used as new training data for the value model. Through comprehensive experiments, we further reveal that both the policy and the critic can be continuously improved for multiple iterations of self-training (Table 2), achieving significant enhancement in search accuracy under reasonable token consumption (Figure 2).\n\nIn addition, we will add the key differences between TS-LLM and our work in Figure 6 in our revision.\n\n```\nW2: Concerns on clarity issues.\n```\nWe sincerely apologize for the inadequate arrangement of the sequence of definitions. We will surely modify the sequence definitions introduced according to your suggestion to make our paper more intelligible to readers. Once again, we would like to express sincere apology for our negligence on these details, and we will certainly modify them in the revised manuscript. Thank you for such a meticulous inspection of our paper!\n\n```\nQ1: Question about the effectiveness of SC.\n```\nRegarding the performance comparison between SC and ReST-MCTS$^*$, we provide all the results among them. In fact, in addition to the experiments in Figure 2 showing that ReST-MCTS$^*$ with multiple iterations is significantly better than SC, other experiments also show that ReST-MCTS$^*$ has a significant improvement over SC. For instance,\n\nIn Figure 2, ReST-MCTS$^*$ significantly outperforms SC after multi-iterations, which verifies the effectiveness of self-training algorithm.\n\nIn Table 4, CoT is indeed the CoT-SC setting, we can find that ReST-MCTS$^*$ outperforms CoT-SC on the separate accuracy of most subjects and the final average accuracy.\n\nIn Table 3, we first provide the results of ReST-MCTS$^*$ on GSM8K and MATH500, which are 86.8 and 37.4, respectively, and then calculate the relative improvement of value models (ORM, MS, and ReST-MCTS*) compared to SC.\n\n| Dataset | SC | ORM | \\% Improv. | \n --- | --- | --- | ---\n| GSM8K | 83.9 | 86.2 | 2.74 |\n| |  | MS | \\% Improv. |\n| |  | 87.1 | 3.81 |\n| |  | ReST-MCTS* | \\% Improv. |\n| |  | 86.8 | 3.46 |\n\n\n| Dataset | SC | ORM | \\% Improv. | \n --- | --- | --- | ---\n| MATH500 | 35.1 | 36.4 | 3.70 |\n| |  | MS | \\% Improv. |\n| |  | 37.3 | 6.27 |\n| |  | ReST-MCTS* | \\% Improv. |\n| |  | 37.4 | 6.55 |\n\nIn the whole view, ReST-MCTS$^*$ significantly outperforms SC in multiple experimental settings, e.g., multi-iterations, scientific reasoning benchmark SciBench, and mathematical reasoning benchmarks.\n\nOnce again, we sincerely thank you for your thoughtful evaluation and valuable suggestions, which have greatly contributed to improving our work. We believe that the revisions we have made adequately address your concerns and questions regarding the comparison and effectiveness of SC. If you believe that our responses have satisfactorily addressed your concerns about the issues, we kindly request that you consider adjusting the final evaluation to reflect this.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JtcxpGRbTV", "reviewer": "Reviewer_rDpc", "summary": "This paper introduces a novel approach for self-training large language models (LLMs) called ReST-MCTS*. This method integrates process reward guidance with Monte Carlo Tree Search (MCTS) to collect high-quality reasoning traces. These traces are then used to train policy and reward models without relying on manual annotations for every reasoning step. The paper claims that this method outperforms existing self-training techniques in terms of accuracy on several reasoning tasks.", "questions": "1. How does the method avoid the scenario where an incorrect intermediate step leads to the correct final answer as mentioned in the paper? Can the Process Reward Model (PRM) alone handle this issue? If an incorrect step during the expansion phase eventually leads to the correct result, would this negatively impact the value function training?\n2. Why is the weighted value designed in its current form? Is there any theoretical justification or practical benefits of this design?\n3. In the LLaMA-3-8B-Instruct results, is the performance of ReSTEM in the first iteration (3.84) a typographical error? Why is the performance so low?", "limitations": "1. Lack of comparison or missing citation of important baseline.\n2. Some design details and implementations can be explained in more detail.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The application of MCTS to improve the capabilities of LLMs is a highly promising approach. This integration allows for more structured and effective exploration of reasoning paths, leading to better model performance.\n2. The proposed method has demonstrated its effectiveness across various reasoning tasks, significantly enhancing the performance of LLMs.", "weaknesses": "1. One major issue with the paper is the lack of novelty, particularly due to the absence of comparison with AlphaLLM [1]. AlphaLLM also uses MCTS to enhance LLM performance, and the ideas and implementation in both papers are strikingly similar. The omission of a comparison or even a citation significantly undermines the contribution and novelty of this paper.\n2. The method still treats each step's reward equally concerning the final answer, only decreasing based on the distance from the root node. This approach does not effectively differentiate the importance of various steps, failing to allocate different weights to different steps appropriately.\n\nReference:\n[1]. Toward Self-Improvement of LLMs via Imagination, Searching, and Criticizing. https://arxiv.org/abs/2404.12253", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable feedback and concerns regarding novelty, the design of reward/value/PRM, and related aspects of our paper. We genuinely appreciate your dedicated time and effort to thoroughly assess our work. We have carefully considered your comments and have made the necessary responses to address these concerns and improve the transparency and credibility of our research. Below, we provide a detailed response to each of your points.\n \n```\nW1 & L1: Concerns on lack of comparison with AlphaLLM.\n```\n**The concurrent work AlphaLLM appeared on ArXiv on April 18th.** We appreciate you bringing this to our attention and will cite AlphaLLM in our paper. \n\nAs an approach that aims to enhance LLM inference, AlphaLLM utilizes a tailored MCTS algorithm and critic models to provide precise feedback. Even though AlphaLLM also adopts MCTS and critic models for self-improvement, their approach is different from ours in various crucial aspects, as elaborated below.\n \n**1) Design of MCTS algorithm.**\nFor the level of search, AlphaLLM’s $\\eta$MCTS considers options as action, with termination signals delivered by a termination function $\\beta$. In contrast, we use reasoning steps as action, which is achieved through tailored prompt design. \n\nConcerning critic models, we use a single value model to provide evaluation for intermediate nodes. The model is trained to predict specially designed quality values that reflect completeness and correctness of partial solutions, rather than estimating the conventional definition of value function in RL. \n\nIn addition, we also incorporate self-critic mechanisms into the tree search algorithm to provide insights for the policy (Appendix C.1), which AlphaLLM does not adopt.\n \n**2) Definition of reward/value.**\nOur definition of weighted reward and quality value is novel, leading to significant differences between our method and AlphaLLM across various processes such as critic model training, data synthesizing, and data filtration. Since our design of quality value involves information on process reward and reasoning distance, our value model trained on this target can naturally provide sufficient feedback during the search, with no need for implementing other critic models mentioned by AlphaLLM.\n \n**3) Self-Training algorithm.**\nAlthough AlphaLLM also includes iterative self-training, the implementation method varies greatly. Most importantly, their critical model is static throughout the iterations, which means they focus more on the improvement of policy. In comparison, we also consider the impacts of self-training on the critic value model. \n\nAs demonstrated in Algorithm 1, we calculate process rewards and quality values according to the final search tree of questions within each iteration, which are then used as new training data for the value model. \n \n```\nW2 & Q2 & L2: Concerns and questions on design of weighted reward/quality value.\n```\nWe appreciate the reviewer's concern and query about the validity and effectiveness of our reward/value design. Please allow us to elaborate more on this issue. We design the weighted reward and quality value this way (Equation 1, 2) for two main reasons. Please see our **Official Comment** for more details on this part.\n \n1) First, we believe that **different reasoning steps should receive varied rewards according to the contribution they make.** To achieve this, we incorporate the process reward $r_{s_k}$ (evaluates the probability of correctness) and the reasoning distance $m_k$ (evaluates contribution or importance, though in an indirect way). These factors involve information that can help evaluate a step more accurately, improving the effectiveness of our method. \n\nW2: Actually, each step’s reward is equal only when the corresponding reasoning trace is the idealized “perfect” solution, where all steps are important, necessary, and concise. Otherwise, steps that make more contributions to the solution have a smaller reasoning distance and higher quality value.\n\n2) Another important reason is scalability and accessibility. Unlike some methods, ours can easily obtain an estimation of each node’s reasoning distance as illustrated in Section 3.2. This enables scaling up our self-training process, which is a significant advantage and practical benefit of our design. Furthermore, Table 2 proves that our design is indeed effective among various tasks and LLMs, further verifying the scalability of our method. We believe this answers the reviewer’s Q2.\n \n```\nQ1: Concerns on the impact of incorrect intermediate steps.\n```\nWe agree that an incorrect intermediate step that reaches the correct answer will be harmful to the iterative training of the value model. However, the possibility of the occurrence of this issue in our work is very small. Our value model is firstly trained on a credible data set that instructs the model to distinguish correct intermediate steps from false ones before self-training. Therefore, during self-training, the MCTS* algorithm can avoid expanding false nodes naturally by probability. \n\nAnother measure for this issue is to filter out the nodes on a trace that reach the correct answer but obtain lower quality values than their parent node (this means they have a negative reward). This helps to purify the newly generated train data based on previous knowledge of the PRM itself. \n \n```\nQ3: Question on the low performance of LLaMA-3-8B-Instruct.\n```\nThank you for raising this question. We have thoroughly reviewed and re-evaluated the results multiple times to ensure the accuracy and reliability of the final result. The result should be 30.84 and we will update this typo.\n \nOnce again, we sincerely thank you for your thoughtful evaluation and valuable suggestions, which have greatly contributed to improving our work. If you believe that our responses have satisfactorily addressed your concerns about the issues, we kindly request that you consider adjusting the final evaluation to reflect this.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "eEgo8uYOhB", "reviewer": "Reviewer_r9nL", "summary": "This paper proposes a novel approach for self-training large language models (LLMs) that combines process reward guidance with Monte Carlo Tree Search (MCTS). This method generates high-quality reasoning traces and per-step values to train policy and reward models, eliminating the need for manual annotation. Experimental validation on multiple benchmarks shows that ReST-MCTS* outperforms existing self-training methods by achieving higher accuracy in reasoning tasks.", "questions": "- Typically, simultaneously training the reward and policy can lead to instability and convergence difficulties. Do you have any concerns about this issue, and how do you address it? Are there any techniques you employ to mitigate these challenges?\n\n- The notation is confused. V_{\\theta} in line 149 is value function but in line 175 is process reward model.", "limitations": "The experiments were conducted on a restricted set of benchmarks, which limits the generalizability of the findings. \n\nAs mentioned in the weaknesses, the scalability of the proposed approach in handling extremely large datasets or more complex reasoning tasks remains uncertain. Addressing this issue with additional strategies or evidence would strengthen the paper and its applicability.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The combination of Monte Carlo Tree Search (MCTS) with process reward models represents a novel method for improving self-training in LLMs. This integration allows for automatic generation of high-quality reasoning traces, which is a significant advancement over existing methods.\n- The paper provides clear definitions and theoretical support for key concepts such as quality value and weighted reward. This enhances its robustness and effectiveness in the self-training process for reasoning problem.", "weaknesses": "- This paper introduces a method to evaluate values for each intermediate step in the reasoning process. Providing more evidence to demonstrate the validity and reasonableness of these intermediate values would make the paper more convincing.\n- Scalability. While the method aims to eliminate manual annotation, the scalability of the proposed approach in extremely large datasets or more complex reasoning tasks might still be a challenge. Maybe complex task require more intermediate steps and wonder if proposed quality value still work for those cases. Providing additional strategies or evidence to support scalability would strengthen the paper.\n- Although the paper demonstrates improved performance on selected benchmarks, a broader range of datasets and tasks would provide a more comprehensive validation of the method’s generalizability and robustness.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for acknowledging our contribution to LLM self-training, clear definitions, and theoretical support and raising valuable concerns and questions about various aspects of our work. We appreciate your dedicated time and effort to thoroughly assess our work. We provide a detailed response to each of them to address your concerns and questions.\n\n```\nW1: Concerns on validity and reasonableness of proposed reward/value design in ReST-MCTS*.\n```\nWe appreciate the reviewer's attention to this concern. Allow us to delve deeper into this concern. The formulation of our weighted reward and quality value (as depicted in Equations 1 and 2) is structured with a specific rationale in mind.\n\nIn our methodology, **we assign varying rewards to different reasoning steps based on their contributions**. This contrasts with previous approaches like Math-Shepherd, which often rely on sparse rewards and the probability of success as values, neglecting the nuanced importance of each step and potentially leading to suboptimal search outcomes.\n\nOur approach integrates the concept of process reward $r_{s_k}$ (assessing the probability of correctness) and reasoning distance $m_k$ (evaluating contribution or importance indirectly). By incorporating these factors into our reward and value design, we aim to enhance the accuracy of step evaluation. The effectiveness of ReST-MCTS$^*$ over MS, as demonstrated in Table 3, underscores the practical benefits and advantages of our design.\n\nFurthermore, through extensive experiments detailed in Table 2, we showcase the efficacy of our design across various tasks and LLMs, thereby verifying the scalability and robustness of our method.\n\n```\nW2 & L2: Concerns on scalability.\n```\nWhile this article mainly addresses mathematical reasoning tasks and shows effectiveness, it is still suitable for working with very large data sets or more complex tasks, such as code generation scenarios, which require additional intermediate steps due to the need for them. To enhance scalability, additional strategies can be explored. For instance,\n\n(1) for longer and more complex codes, higher-level structures can be used as a single reasoning step. These structures could include lines of code, code blocks, or even entire functions.\n\n(2) similar to approaches used in math reasoning tasks, a similar PRM provides expert reward feedback for value function estimation. In this way, the MCTS algorithm can always filter out better traces regardless of the policy’s competence. Alternatively, if we continue to rely on MC simulations, the policy and an existing reward model could be updated during this process. RL methods such as Q-learning can be adopted.\n\n```\nW3 & L1: Concerns on a broader range of datasets and tasks.\n```\nTo further bolster the method's generalizability and robustness, expanding the evaluation to encompass a broader range of datasets and tasks beyond common mathematical reasoning could offer a more comprehensive validation. \n\nAs highlighted previously, this study primarily concentrates on mathematical reasoning tasks, which is reflected in the evaluation of mathematical benchmarks such as MATH (Table 2, Figure 2), GSM8K (Table 3), MATH500 (Table 3), and various Math subsets (Table 4).\n\nMoreover, the method's performance is also assessed on scientific reasoning benchmarks, including GPQA_{Diamond} and CEval-Hard (Table 2), as well as SciBench datasets covering Mathematics, Chemistry, and Physics (Table 4, Table 6, Figure 5), and SciEval (Table 7).\n\nCorresponding to the W2 & L2, this paper provides a more comprehensive assessment of the method's effectiveness across a wider spectrum of applications and facilitates a deeper understanding of its generalizability. \n\n```\nQ1: Concerns about simultaneously training the reward and policy.\n```\nWe acknowledge the importance of considering the issue of online RL. However, **it is vital to emphasize that our methodology primarily focuses on the offline self-training paradigm**. Within this framework, data is generated via MCTS$^*$ using a static policy and critic within each iteration. The newly synthesized data is verified and filtered according to the ground truth rather than the outputs of the value model. Subsequently, this filtered data is utilized to individually train the policy and value models, aligning our algorithm with an offline and relatively stable approach.\n\nIn tackling the instability and convergence challenges inherent in simultaneously training the reward and policy and enhancing the overall performance and robustness of the training process, typical of online RL strategies, we can employ several techniques:\n\nExperience Replay: firstly, during training, the agent stores transition in a replay buffer, accumulating experiences over time. Then, instead of using experiences immediately, the agent samples mini-batches of experiences randomly from the replay buffer during training. By sampling randomly from past experiences, experience replay breaks the temporal correlations present in sequential data, which can help prevent the model from getting biased toward recent experiences. Finally, using experience replay can lead to more stable and efficient learning by providing a diverse set of experiences for the agent to learn from, smoothing out the learning process.\n\n```\nQ2: Concerns on the $V_\\theta$.\n```\nThanks for the suggestion, $V_{\\theta}$ is indeed the definition of the process reward model. We will update this typo (line 149) in our manuscript.\n\nOnce again, we sincerely thank you for your thoughtful evaluation and valuable suggestions, which have greatly contributed to improving our work. We believe that the revisions we have made adequately address your concerns and questions regarding quality value, scalability, and related aspects of ReST-MCTS*. If you believe that our responses have satisfactorily addressed your concerns about the issues, we kindly request that you consider adjusting the final evaluation to reflect this.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nj58PoMvO2", "reviewer": "Reviewer_S8hx", "summary": "This paper proposed ReST-MCTS* to assist the large language model to answer reasoning questions. A variant of MCTS, which utilizes the evaluation of current state as the value function, is employed to \nautomatically annotate the process reward of each intermediate node via sufficient times of rollouts. A self-refine process is employed to finetune the LLMs. Tested on SciBench, ReST-MCTS* has achieved better performance than existing self-training approaches and reasoning policies.", "questions": "1. Do different algorithms have differences in their actual running time, considering that MCTS requires a significant amount of Monte Carlo simulations to estimate action values, often resulting in higher computational requirements.\n\n2. The definition of $r_{s_k}$ needs to be provided in advance. It is mentioned in Line 135, but is defined in Line 181. If I do not know that $r_{s_k}=1-r_{s_k}^{HE}$, observation 2 will be confusing to me.\n\n3. Why is the assumption made that $w_{s_k}\\in [0,1]$ when proving Theorem 1 in Appendix C.1? It is obvious that $w_{s_k}$ can be negative.  \n\n4. In line 648-650, $v_{k-1}\\in [0,1]$ and $v_k\\in [v_{k-1}, 1+v_{k-1}]$, why this makes $v_k$ in the range of $[0,1]$. It seems $v_k$ is in $[0,2]$.", "limitations": "Please refer to the Weaknesses and Questions part.", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper builds a value and policy network to assist the reasoning process of LLM , borrowing from the MuZero framework, and has achieved siginicant improvements.\n\n2. Theoratical analysis is given to help the understanding of ReST-MCTS* algorithm.\n\n3. The experiment is conducted comprehensively and convincingly.\n\n4. The paper is written concisely and clearly.", "weaknesses": "1. More discussion about the value function is needed. In refincement learning, value is the expected total reward, rather than the evaluation of the current state. These two definitions are completely different, and I would like to see more explanations. For example:\n\n    a) Can UCT still converge to the optimal solution when the simulation times approach infinity?\n    \n    b) In most cases, $v_k=v_{k-1}+w_{s_k}$ is established for two neighboring nodes in the search tree. Assume $S^k=\\\\{s_1^k,s_2^k,\\cdots, s_N^k\\\\}$ denotes all children nodes of $s_{k-1}$. All nodes in $S^k$ share the same $v_{k-1}$ value in Equation (1). $v_{k-1}$ and $m_k$ are also the same in Equation (2), only $r_{s_k}$ is different. Considering that MCTS select the traversed node from the set of nodes with the same parent node while do simulation, why not predict $r_{s_k}$ directly?\n\n    c) In the value backpropagation, $v_C$ of $s_t$ seems to be the weighted sum of the estimated values of all expanded nodes in the subtree rooted with $s_t$. What's the meaning of the average of state evaluation values? A larger evaluation value does not necessarily indicate a better expansion direction, especially when the distance to the goal is far.\n\n    d) Using evaluation score as the value function makes the algorithm more likely to be a multi-step greedy algorithm, instead of a heuristic search algorithm like MCTS. What is your opinion on this matter?\n\n2. Can you provide evidence of the quality of value function training based on prediction errors, in addition to the search results? Sometimes, even if the search results are good, it doesn't necessarily mean that the value function predictions are accurate. For example, set the estimated values to always be $0$, $A^*$ search can also perform well in some situations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "​​Thank you for acknowledging our contribution to LLM reasoning and raising valuable concerns and questions about various aspects of our work. We appreciate the time and effort you have dedicated to thoroughly assessing our work. To address your concerns and questions, we now provide a detailed response to each of them.\n```\nQ2 & Q3 & Q4: Questions about specific definitions, assumptions, and conclusions.\n```\nWe want to clarify these questions first to help better understand this work. For Q2, we will surely rearrange the sequence of definitions as suggested. For Q3 and Q4 which focus on the deduction details in Appendix B.1, we sincerely apologize for making a mistake. In fact, reaching the conclusion of $v_k \\in [0,1]$ does not require the assumption of $w_{s_k} \\in [0,1]$. As an important factor measuring the correctness and contribution of a reasoning step, $w_{s_k}$ is designed to be signed and bounded within the range $[-1,1]$, rather than $[0,1]$. We’ve demonstrated the details in the **Official Comment**.\n \n```\nW1: Concerns on the definition of quality value.\n```\nWe appreciate your concerns about the design of weighted reward and quality value. They remind us that there are a few things we may have neglected and require more explanation. Please refer to the **Global Author Rebuttal** for a general explanation of our approach design.\n \na) **Your concerns regarding the convergence of UCT for our case are profound and thoughtful.** As mentioned in the **Global Author Rebuttal**, our methodology is offline RL and doesn’t face issues with UCT convergence. However, we agree that the convergence of UCT in our setting for online RL is unclear. In consideration that our definitions of reward and value are novel, it certainly requires considerable effort to design an appropriate online RL paradigm that aligns with these definitions. \n \nb) In fact, under our definition, the reasoning distance $m_k$ is the minimum reasoning steps required to access the final answer starting from a node with partial solution $p_k=[s_1, s_2, \\cdots, s_k]$. This means when we generate children nodes of $s_{k-1}$, the reasoning distance of these nodes, denoted as $m_k^i(i=1, 2, \\cdots, N)$, is already different. Different children make different contributions to the final answer, leading to varied search directions that require different numbers of reasoning steps. Thus, **though $v_{k-1}$ is the same, both $r_{s_k}$ and $m_k$ are different**.\n \nc) The value backpropagation process is not compulsory. However, we regard this process as a heuristic that provides insight into the selection of search direction. Updating value estimation using the average method promotes exploitation in more promising directions in general.  Please see the details in our **Official Comment**.\n\nd) We believe former discussions have addressed this question. By adopting UCT and value backpropagation, our algorithm values exploration as important as exploitation. Concerning implementation, the exploration constant also allows the different extents of the exploration. Though we use a different value definition, the core idea of MCTS has not changed.\n \n```\nW2: Concerns on the quality of the value model.\n```\nWe appreciate your concerns that our value model may not be truly effective. To address your concern, we present two pieces of evidence aside from the results of the search. \n\nFirstly, our value model achieves an accuracy of approximately 70% with an absolute tolerance of 0.1 on the test set consisting of 14k data samples, as demonstrated in Appendix D.1. This means the value model can already gain considerable knowledge even before self-training, justifying its quality. \n\nMoreover, we also experiment to compare the performances of critics alone. We use the same policy and the same CoT sampling strategy to generate solutions for GSM8K and MATH500. These solutions are then evaluated and filtered using different methods. Results shown in Table 3 indicate that the filtration method based on our value model + SC achieves the best accuracy on both datasets, outperforming baselines like SC, ORM, and PRM of Math-Shepherd. This also justifies the quality of our value model, since all other influences are eliminated.\n \n```\nQ1: Questions on running time and computational requirements of different algorithms.\n```\nConcerning computational costs, we have already compared the number of tokens consumed by each algorithm to achieve certain accuracy on MATH and SciBench, as shown in Figure 2. Results reveal that to reach a certain expected accuracy, MCTS* basically requires fewer tokens than other algorithms like SC and ORM+BoN. This means that based on the same expected standards, MCTS$^*$ outperforms other algorithms while maintaining a reasonable computation cost. As for the actual running time, we have recorded the average running time of different algorithms (under our basic experiment settings) on a single question, as shown below.\n\n| Method | CoT+SC | ORM+BoN | PRM+BoN | MCTS* | \n --- | --- | --- | ---| ---\n| Running time (s) | 41 | 43 | 73 | 108 |\n| Accuracy on MATH (%) | 37.0 | 33.5 | 34.5 | 41.5 |\n\nIndeed, MCTS$^*$ spends more time on exploration and simulation than other simple algorithms. However, since our method adopts a different design of value, it doesn’t require massive Monte Carlo estimations. This reduces the running time of our algorithm and limits the time consumption to a reasonable range. Notice that MCTS* can achieve high accuracy that other algorithms can never attain even at unlimited cost, we believe this extra time is fairly acceptable.\n \nLastly, we kindly appreciate your in-depth evaluation of our work and thank you for your valuable questions and suggestions, which have greatly contributed to improving our work. If you believe that our responses have satisfactorily addressed your concerns about the issues, we kindly request that you consider adjusting the final evaluation to reflect this.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear ACs and reviewers,

thank you very much for your valuable feedback. We list the main issues raised by reviewers and explain them below.

```
The motivation and reason behind the design of ReST-MCTS*.
```
First, the main reason **we design a different reward and value for MCTS lies in the defect of conventional reward/value for the case of reasoning**. We believe that different reasoning steps (actions) should receive varied rewards according to the contribution they make, while conventional methods like Math-Shepherd often consider using sparse reward (only the final step receives reward) and probability of success as value. These methods omit the varied importance of steps and can lead to ineffective searches. On the other hand, modeling intrinsic rewards that follow the conventional RL paradigm is very difficult. Some approaches like AlphaLLM attempt to model the reward of intermediate steps in a descriptive way, but its implementation is complicated and the results are still unsatisfactory. These issues inspire us to look for alternative reward and value designs that may be more suitable for reasoning tasks, jumping out from the conventional RL paradigm. 

```
The benefits of the design of ReST-MCTS*.
```
The design of the proposed ReST-MCTS$^*$ has three main benefits: **accurate evaluation of intermediate steps, balanced exploration and exploitation, and scalability**. 

Firstly, we achieve a better evaluation of steps by incorporating the process reward $r_{s_k}$  (evaluates the probability of correctness of a single step) and the reasoning distance $m_k$ (evaluates contribution or importance, though in an indirect way). These factors involve information that can help assess a step more accurately, thus we believe involving these factors in the design of reward and value may improve the effectiveness of our method. 

Secondly, we regard MCTS as a heuristic algorithm that balances exploration and exploitation, since it considers both visit frequency and expected reward in the UCT formula. Despite the differences in reward/value definition, we adopt the core UCT formula to achieve a balanced search in MCTS$^*$, resulting in better search outcomes.

Thirdly, our design enables easy acquisition of estimations of each node’s reasoning distance and quality value as illustrated in Section 3.2. This enables scaling up our self-training process, which is a significant advantage and practical benefit of our design. Furthermore, Table 2 proves that ReST-MCTS$^*$ is indeed effective among various tasks and LLMs, further verifying the scalability of our method. 

```
Different definitions of value functions in traditional RL and our proposed MCTS*.
```
Compared to the expected total reward in traditional RL, our value function focuses on the evaluation of the current state. Even though the settings of reward/value are different in our work, the core of the algorithm can still be preserved and adopted. On one side, we want to further explore the nodes that make more progress in the right direction (higher quality value). On the other side, we can not simply discard the search directions that haven’t made much progress for some reason. As a special search algorithm, MCTS$^*$ inherits the core decision method of MCTS and adapts to the usage of weighted reward and quality value. It suffices to perform scalable data synthesis for iterative self-training of policy and value models, achieving significant improvements as our experiments show.

```
ReST-MCTS* focuses on the offline self-training paradigm.
```
It is crucial to emphasize that our methodology predominantly focuses on the offline self-training paradigm. Within this framework, data is generated through MCTS$^*$ using a static policy and critic within each iteration, as elaborated in Section 3.2. The newly synthesized data is verified and filtered according to the ground truth, rather than the outputs of the value model. Subsequently, this filtered data is utilized to individually train the policy and value models, rendering our algorithm offline and relatively stable.

Meta-review: Despite raising several concerns regarding soundness of the weighted reward approach, clarity, positioning within existing literature, and evaluation, the authors thoroughly addressed these concerns and all reviewers recommended acceptance with varying strength. I will therefore join them in recommending acceptance, but will note that, having read the discussion, I am not satisfied beyond all doubt that the weighted reward approach is indeed sound, and would encourage the authors to justify it more thoroughly. I'd also like to note that unpublished works like AlphaLLM, which seem to have only been posted on arXiv with the footnote “Work in progress”, should not be considered prior work for peer reviewed publications, although the authors would do well to acknowledge them.
Decision: Accept (poster)