Improving mathematical reasoning with process supervision

Let’s Verify Step by Step
Hunter Lightman∗Vineet Kosaraju∗Yura Burda∗Harri Edwards
Bowen Baker Teddy Lee Jan Leike John Schulman Ilya Sutskever
Karl Cobbe∗
OpenAI
Abstract
In recent years, large language models have greatly improved in their
ability to perform complex multi-step reasoning. However, even state-
of-the-art models still regularly produce logical mistakes. To train more
reliable models, we can turn either to outcome supervision, which provides
feedback for a final result, or process supervision, which provides feedback
for each intermediate reasoning step. Given the importance of training
reliable models, and given the high cost of human feedback, it is impor-
tant to carefully compare the both methods. Recent work has already
begun this comparison, but many questions still remain. We conduct our
own investigation, finding that process supervision significantly outper-
forms outcome supervision for training models to solve problems from the
challenging MATH dataset. Our process-supervised model solves 78% of
problems from a representative subset of the MATH test set. Additionally,
we show that active learning significantly improves the efficacy of process
supervision. To support related research, we also release PRM800K, the
complete dataset of 800,000 step-level human feedback labels used to train
our best reward model.
1 Introduction
Large language models are capable of solving tasks that require complex multi-
step reasoning by generating solutions in a step-by-step chain-of-thought format
(Nye et al., 2021; Wei et al., 2022; Kojima et al., 2022). However, even state-
of-the-art models are prone to producing falsehoods — they exhibit a tendency
to invent facts in moments of uncertainty (Bubeck et al., 2023). These hallu-
cinations (Maynez et al., 2020) are particularly problematic in domains that
require multi-step reasoning, since a single logical error is enough to derail a
much larger solution. Detecting and mitigating hallucinations is essential to
improve reasoning capabilities.
∗Primary authors. Correspondence to: Karl Cobbe <karl@openai.com >
1arXiv:2305.20050v1  [cs.LG]  31 May 2023
One effective method involves training reward models to discriminate be-
tween desirable and undesirable outputs. The reward model can then be used
in a reinforcement learning pipeline (Ziegler et al., 2019; Stiennon et al., 2020;
Nakano et al., 2021; Ouyang et al., 2022) or to perform search via rejection sam-
pling (Nichols et al., 2020; Shen et al., 2021; Cobbe et al., 2021). While these
techniques are useful, the resulting system is only as reliable as the reward
model itself. It is therefore important that we study how to most effectively
train reliable reward models.
In closely related work, Uesato et al. (2022) describe two distinct meth-
ods for training reward models: outcome supervision and process supervision.
Outcome-supervised reward models (ORMs) are trained using only the final
result of the model’s chain-of-thought, while process-supervised reward models
(PRMs) receive feedback for each step in the chain-of-thought. There are com-
pelling reasons to favor process supervision. It provides more precise feedback,
since it specifies the exact location of any errors that occur. It also has sev-
eral advantages relevant to AI alignment: it is easier for humans to interpret,
and it more directly rewards models for following a human-endorsed chain-of-
thought. Within the domain of logical reasoning, models trained with outcome
supervision regularly use incorrect reasoning to reach the correct final answer
(Zelikman et al., 2022; Creswell et al., 2022). Process supervision has been
shown to mitigate this misaligned behavior (Uesato et al., 2022).
Despite these advantages, Uesato et al. (2022) found that outcome supervi-
sion and process supervision led to similar final performance in the domain of
grade school math. We conduct our own detailed comparison of outcome and
process supervision, with three main differences: we use a more capable base
model, we use significantly more human feedback, and we train and test on the
more challenging MATH dataset (Hendrycks et al., 2021).
Our main contributions are as follows:
1. We show that process supervision can train much more reliable reward
models than outcome supervision. We use our state-of-the-art PRM to
solve 78 .2% of problems from a representative subset of the MATH test
set.
2. We show that a large reward model can reliably approximate human su-
pervision for smaller reward models, and that it can be used to efficiently
conduct large-scale data collection ablations.
3. We show that active learning leads to a 2 .6×improvement in the data
efficiency of process supervision.
4. We release our full process supervision dataset, PRM800K, to promote
related research.
2
2 Methods
We perform a comparison of outcome and process supervision, following a sim-
ilar methodology to Uesato et al. (2022). Outcome supervision can be provided
without humans, since all problems in the MATH dataset have automatically
checkable answers. In contrast, there is no simple way to automate process su-
pervision. We therefore rely on human data-labelers to provide process super-
vision, specifically by labelling the correctness of each step in model-generated
solutions.
We conduct experiments in two separate regimes: large-scale and small-
scale. Each has its own advantages, and they offer complimentary perspectives.
At large-scale, we finetune all models from GPT-4 (OpenAI, 2023). We focus
on advancing the state-of-the-art by training the most reliable ORM and PRM
possible. Unfortunately the training sets for these reward models are not directly
comparable, for reasons we will discuss in Section 3. These models are therefore
not ideal for making an apples-to-apples comparison of outcome and process
supervision. To address this flaw, we also train models at small-scale, where
we can conduct a more direct comparison. In order to remove our dependence
on costly human feedback, we use a large-scale model to supervise small-scale
model training. This setup enables us to conduct several important ablations
that would otherwise be infeasible.
2.1 Scope
At each model scale, we use a single fixed model to generate all solutions. We
call this model the generator . We do not attempt to improve the generator with
reinforcement learning (RL). When we discuss outcome and process supervision,
we are specifically referring to the supervision given to the reward model. We do
not discuss any supervision the generator would receive from the reward model
if trained with RL. Although finetuning the generator with RL is a natural next
step, it is intentionally not the focus of this work.
We instead focus exclusively on how to train the most reliable reward model
possible. We evaluate a reward model by its ability to perform best-of-N search
over uniformly sampled solutions from the generator. For each test problem we
select the solution ranked highest by the reward model, automatically grade it
based on its final answer, and report the fraction that are correct. A reward
model that is more reliable will select the correct solution more often.
2.2 Base Models
All large-scale models are finetuned from the base GPT-4 model (OpenAI, 2023).
This model has been pretrained solely to predict the next token; it has not been
pretrained with any Reinforcement Learning from Human Feedback (RLHF)
(Christiano et al., 2017). The small-scale base models are similar in design to
GPT-4, but they were pretrained with roughly 200 times less compute. As an
additional pretraining step, we finetune all models on a dataset of roughly 1.5B
3
Figure 1: A screenshot of the interface used to collect feedback for each step in
a solution.
math-relevant tokens, which we call MathMix. Similar to Lewkowycz et al.
(2022), we find that this improves the model’s mathematical reasoning capabil-
ities. Details on how this dataset was constructed can be found in Appendix A.
2.3 Generator
To make parsing individual steps easier, we train the generator to produce
solutions in a newline delimited step-by-step format. Specifically, we few-shot
generate solutions to MATH training problems, filter to those that reach the
correct final answer, and finetune the base model on this dataset for a single
epoch. This step is not intended to teach the generator new skills; it is intended
only to teach the generator to produce solutions in the desired format.
2.4 Data Collection
To collect process supervision data, we present human data-labelers with step-
by-step solutions to MATH problems sampled by the large-scale generator.
Their task is to assign each step in the solution a label of positive ,negative ,
orneutral , as shown in Figure 1. A positive label indicates that the step is cor-
rect and reasonable. A negative label indicates that the step is either incorrect
or unreasonable. A neutral label indicates ambiguity. In practice, a step may
be labelled neutral if it is subtly misleading, or if it is a poor suggestion that
is technically still valid. We permit neutral labels since this allows us to defer
the decision about how to handle ambiguity: at test time, we can treat neutral
labels as either positive or negative. A more detailed description of the labelling
instructions is provided in Appendix D.
We label solutions exclusively from the large-scale generator in order to
maximize the value of our limited human-data resource. We refer to the en-
tire dataset of step-level labels collected as PRM800K. The PRM800K training
set contains 800K step-level labels across 75K solutions to 12K problems. To
4
minimize overfitting, we include data from 4.5K MATH test problems in the
PRM800K training set, and we therefore evaluate our models only on the re-
maining 500 MATH test problems. More details about this test set can be found
in Appendix C.
During data collection, we must decide which solutions to surface to data-
labelers. The most straightforward strategy is to uniformly surface solutions
produced by the generator. However, if we surface solutions that make obvious
errors, the human feedback we get is less valuable. We would prefer to surface
solutions that are more likely to fool our best reward model. To that end, we at-
tempt to strategically select which solutions to show data-labelers. Specifically,
we choose to surface convincing wrong-answer solutions. We use the term con-
vincing to refer to solutions that are rated highly by our current best PRM, and
we use wrong-answer to refer to solutions that reach an incorrect final answer.
We use this slightly verbose phrasing to emphasize the fact that correctness is
determined solely by checking the final answer, a process which occasionally
leads to misgraded solutions. We expect to gain more information from labeling
convincing wrong-answer solutions, since we know the PRM is mistaken about
at least one step in each such solution.
In addition to using this selection strategy, we also iteratively re-train our
PRM using the latest data at several points in the data collection process. At
each iteration, we generate N solutions per problem and surface only the top K
most convincing wrong-answer solutions to data-labelers. We experiment with
either applying this top-K filtering at a problem level (K solutions per problem)
or globally across the dataset (K solutions in total, unequally distributed among
problems). Since the data collection process is expensive, it was not feasible
to conduct at-scale ablations of these decisions. However, we perform several
surrogate ablations in Section 4, using our largest PRM as a labelling oracle for
a smaller PRM. More details about data collection can be found in Appendix B.
2.5 Outcome-supervised Reward Models (ORMs)
We train ORMs following a similar methodology to Cobbe et al. (2021). We
uniformly sample a fixed number of solutions per problem from the generator,
and we train the ORM to predict whether each solution is correct or incorrect.
In practice, we usually determine correctness by automatically checking the
final answer, but in principle these labels could be provided by humans. At test
time, we use the ORM’s prediction at the final token as the overall score for the
solution. We note the automatic grading used to determine ORM targets is not
perfectly reliable: false positives solutions that reach the correct answer with
incorrect reasoning will be misgraded. We discuss additional ORM training
details in Appendix E.
2.6 Process-supervised Reward Models (PRMs)
We train PRMs to predict the correctness of each step after the last token in
each step. This prediction takes the form of a single token, and we maximize the
5
Figure 2: Two solutions to the same problem, graded by the PRM. The solution
on the left is correct while the solution on the right is incorrect. A green
background indicates a high PRM score, and a red background indicates a low
score. The PRM correctly identifies the mistake in the incorrect solution.
log-likelihood of these target tokens during training. The PRM can therefore
be trained in a standard language model pipeline without any special accom-
modations. To determine the step-level predictions at test time, it suffices to
perform a single PRM forward pass over the whole solution. We visualize large-
scale PRM scores for two different solutions in Figure 2. To compare multiple
solutions, it is necessary to compute a single score for each solution. This is an
important but straightforward detail: we define the PRM score for a solution to
be the probability that every step is correct under the PRM. We implement this
as the product of the correctness probabilities for each step. We describe other
possible scoring strategies and additional PRM training details in Appendix F.
When we provide process supervision, we deliberately choose to supervise
only up to the first incorrect step. This makes the comparison between out-
come and process supervision more straightforward. For correct solutions, both
methods provide the same information, namely that every step is correct. For
incorrect solutions, both methods reveal the existence of at least one mistake,
and process supervision additionally reveals the precise location of that mistake.
If we were to provide additional process supervision beyond the first mistake,
then process supervision would have an even greater information advantage.
This decision also keeps the labelling cost similar for humans: without relying
on an easy-to-check final answer, determining the correctness of a solution is
equivalent to identifying its first mistake. While most MATH problems do have
easy-to-check final answers, we expect this to not remain true in more complex
domains.
6
ORM PRM Majority Voting
% Solved (Best-of-1860) 72.478.2 69.6
101102103
N = number of solutions per problem626466687072747678% Problems Solved (Best-of-N)
Process-Supervised RM
Outcome-Supervised RM
Majority Voting
Figure 3: A comparison of outcome-supervised and process-supervised reward
models, evaluated by their ability to search over many test solutions. Majority
voting is shown as a strong baseline. For N≤1000, we visualize the variance
across many subsamples of the 1860 solutions we generated in total per problem.
3 Large-scale Supervision
We train the large-scale PRM using the step-level labels in PRM800K. To ensure
the large-scale ORM baseline is as strong as possible, we train on 100 uniform
samples per problem from the generator. This means the ORM training set has
no overlap with PRM800K, and it is an order of magnitude larger. Although
these two training sets are not directly comparable, each represents our best
attempt to advance the state-of-the-art with each form of supervision. We note
that training the ORM solely on PRM800K solutions would be problematic,
since our active learning strategy has heavily biased the dataset towards wrong-
answer solutions. We did explore training the ORM on a superset of PRM800K
solutions, by mixing in uniformly sampled solutions, but we found that this did
not improve ORM performance.
Figure 3 shows how the best-of-N performance of each reward model varies
as a function of N. Since majority voting is known to be a strong baseline (Wang
et al., 2022; Lewkowycz et al., 2022), we also include this method as a point of
comparison. While the ORM performs slightly better than the majority voting
baseline, the PRM strongly outperforms both. Not only does the PRM reach
higher performance for all values of N, but the performance gap widens as N
increases. This indicates that the PRM is more effective than both the ORM and
majority voting at searching over a large number of model-generated solutions.
7
100101102
Number of solutions labelled per problem2530354045505560% Problems Solved (Best-of-500)PRM + Active Learning
PRM (PRMlarge supervised)
ORM (PRMlarge supervised)
ORM (final-answer supervised)(a) Four series of reward models
trained using different data collection
strategies, compared across training
sets of varying sizes.
100101102103
N = number of solutions per problem202530354045505560% Problems Solved (Best-of-N)PRM (PRMlarge supervised)
ORM (PRMlarge supervised)
ORM (final-answer supervised)(b) Three reward models trained on
200 samples/problem using different
forms of supervision, compared across
many test-time compute budgets.
Figure 4: A comparison of different forms of outcome and process supervision.
Mean and standard deviation is shown across three seeds.
We experimented with using RM-weighted voting (Li et al., 2022; Uesato et al.,
2022) to combine the benefits of the PRM and majority voting, but this did not
noticeably improve performance. We use a specific subset of the MATH test set
for evaluation, which we describe in Appendix C. We further break down these
results by problem difficulty in Appendix G.
4 Small-scale Synthetic Supervision
We find that the PRM outperforms the ORM at large-scale, but this result alone
paints an incomplete picture. To better compare outcome and process supervi-
sion, there are two confounding factors that must be isolated. First, the training
sets for the ORM and the PRM are not directly comparable: the PRM training
set was constructed using active learning, is biased towards answer-incorrect
solutions, and is an order of magnitude smaller. Second, the final-answer grad-
ing will provide positive labels to spurious solutions that reach the correct final
answer despite incorrect reasoning. This could damage ORM performance, an
effect we may or may not want to attribute to outcome supervision more gen-
erally.
Due to the high cost of collecting human feedback, we cannot easily ablate
these factors using human labelers. We instead perform the relevant ablations
by using the large-scale PRM to supervise smaller models. This setup enables
us to simulate a large amount of data collection at a modest cost. For the
remainder of this section, we refer to the large-scale PRM from Section 3 as
PRM large.
8
4.1 Process vs Outcome Supervision
We now conduct a direct comparison of outcome and process supervision. We
first sample between 1 and 200 solutions per problem from a small-scale genera-
tor. For each dataset, we provide three forms of supervision: process supervision
from PRM large, outcome supervision from PRM large, and outcome supervision
from final-answer checking. The choice of supervision is the only difference
between these three series of reward models, which are otherwise trained on
identical datasets. See Appendix H for more details about how PRM large is
used for outcome and process supervision.
In Figure 4a, we evaluate each reward model by its best-of-500 selection. We
see that process supervision significantly outperforms both forms of outcome
supervision at all data collection scales. In Figure 4b, we evaluate the best
reward model from each series by its best-of-N performance across different
values of N. We see that using PRM large for outcome supervision is noticeably
more effective than final-answer checking. This can be explained by the fact
that PRM large provides better supervision for solutions that reach the correct
final answer using incorrect reasoning.
It is not clear whether supervision by PRM largeor by final-answer checking
represents the more appropriate outcome supervision baseline. While final-
answer supervision is more explicitly outcome based, its main weakness — the
existence of false positives — is arguably over-emphasized in the MATH dataset.
Outcome supervision by PRM largebetter represents outcome supervision in do-
mains that are less susceptible to false positives. We consider outcome supervi-
sion by PRM largeto be the more relevant baseline, but we encourage the reader
to draw their own conclusions.
4.2 Active Learning
Finally, we investigate the impact of active learning. We train a small-scale
reward model, PRM selector , on a single sample from each problem, and we use
this model to score 1000 samples per problem. To train each of our larger re-
ward models, we select Nsamples per problem such that 80% are the most
convincing (according to PRM selector ) wrong-answer samples, and 20% are the
most convincing samples that remain (right- or wrong-answer). We score the
selected samples with PRM largeand train on those scores. This process ensures
that all samples are relatively convincing under PRM selector , that a large frac-
tion are known to contain at least one mistake, and that our overall dataset
is not too heavily biased toward wrong-answer solutions. Performance of this
data labelling scheme is shown in Figure 4a. By comparing the slopes of the
line of best fit with and without active learning, we estimate that this form
of active learning is approximately 2.6x more data efficient than uniform data
labelling. We note that the model trained on the largest active learning dataset
(200 samples per problem) appears to slightly underperform the expected trend
line. Our best explanation for this observation is that 200 samples represents
a significant fraction of the overall selection pool (1000 samples) and that this
9
ORM PRM Majority Vote # Problems
AP Calculus 68 .9% 86.7% 80.0% 45
AP Chemistry 68 .9% 80.0% 71.7% 60
AP Physics 77 .8% 86.7% 82.2% 45
AMC10/12 49 .1% 53.2% 32.8% 84
Aggregate 63 .8% 72.9% 61.3% 234
Table 1: We measure out-of-distribution generalization using recent STEM tests.
We evaluate the outcome-supervised RM, the process-supervised RM, and ma-
jority voting using 100 test samples per problem.
relative lack of diversity limits the possible upside from active learning.
We also performed a preliminary investigation into the impact of iteratively
retraining PRM selector throughout data collection. Between iterations, we re-
trained PRM selector using all currently labeled data. Unfortunately, we observed
instability in this process which we were unable to diagnose. The resulting
reward models performed no better than the models described above. We expect
some form of iterative retraining to be beneficial in active learning, but we
currently have no concrete evidence to support this claim. We consider this a
compelling direction for future research.
5 OOD Generalization
To get some measure of out-of-distribution generalization, we evaluate our large-
scale ORM and PRM on a held-out set of 224 STEM questions, pulled from the
most recent AP Physics, AP Calculus, AP Chemistry, AMC10, and AMC12 ex-
ams. Since these tests were released after the pre-training dataset was compiled,
we can have high confidence that the model has not seen these problems. We
report the best-of-100 performance of the ORM, PRM and majority voting in
Table 1. We observe results similar to those in Section 3: the PRM outperforms
both the ORM and majority voting. This shows us that the PRM can tolerate
a modest amount of distribution shift and that its strong performance holds up
on fresh test questions.
6 Discussion
6.1 Credit Assignment
One clear advantage of process supervision is that it provides more precise
feedback than outcome supervision. A reward model trained with outcome
supervision faces a difficult credit-assignment task — to generalize well, it must
determine where an incorrect solution went wrong. This is particularly difficult
for hard problems: most model-generated solutions contain an error somewhere,
so the marginal value of a negative label from outcome supervision is low. In
10
contrast, process supervision provides a richer signal: it specifies both how
many of the first steps were in fact correct, as well as the precise location of
the incorrect step. Process supervision makes credit assignment easier, and we
believe that this explains its strong performance.
6.2 Alignment Impact
Process supervision has several advantages over outcome supervision related
to AI alignment. Process supervision is more likely to produce interpretable
reasoning, since it encourages models to follow a process endorsed by humans.
Process supervision is also inherently safer: it directly rewards an aligned chain-
of-thought rather than relying on outcomes as a proxy for aligned behavior
(Stuhlm¨ uller and Byun, 2022). In contrast, outcome supervision is harder to
scrutinize, and the preferences conveyed are less precise. In the worst case,
the use of outcomes as an imperfect proxy could lead to models that become
misaligned after learning to exploit the reward signal (Uesato et al., 2022; Cotra,
2022; Everitt et al., 2017).
In some cases, safer methods for AI systems can lead to reduced performance
(Ouyang et al., 2022; Askell et al., 2021), a cost which is known as an alignment
tax. In general, any alignment tax may hinder the adoption of alignment meth-
ods, due to pressure to deploy the most capable model. Our results show that
process supervision in fact incurs a negative alignment tax. This could lead to
increased adoption of process supervision, which we believe would have positive
alignment side-effects. It is unknown how broadly these results will generalize
beyond the domain of math, and we consider it important for future work to
explore the impact of process supervision in other domains.
6.3 Test Set Contamination
The test set of the MATH dataset contains problems that are discussed in
several online venues, and it is likely that some of these problems appear in
the pretraining dataset for our models. We attempted to remove all MATH
problems from our MathMix dataset using string-matching heuristics, but since
humans can post hard-to-detect rephrasings of a problem online, it is difficult
to make any strong guarantees about the overlap between MathMix and the
MATH dataset.
In our experience inspecting model-generated solutions, we saw no clear signs
of our models memorizing MATH problems. However, it is impossible to rule
out subtle forms of memorization that would slip past manual inspection, and
it is still possible that some degree of contamination has slightly inflated our
performance on the MATH test set. Even in that case, we would expect any
contamination to manifest similarly across all methods, and that the relative
comparisons made throughout this work would remain mostly unaffected.
We also note that the PRM regularly surfaces correct solutions to MATH
problems that have a low single-digit percentage solve-rate under the genera-
tor, some examples of which can be seen in Appendix I. The generator’s low
11
solve-rate is an additional indication that it has not encountered such problems
via test set contamination. Our generalization results from Section 5 further
strengthen our claim that test set contamination has not significantly impacted
this work, since we observe qualitatively similar results on problems that are
guaranteed to be uncontaminated.
7 Related Work
7.1 Outcome vs Process Supervision
In work closely related to our own, Uesato et al. (2022) compare the impact
of outcome and process supervision in the domain of grade school math. They
found that both methods led to similar final-answer error rates, and that process
supervision achieved those results with less data. While our core methodology is
very similar, there are three main details that differ. First, we use a more capable
model to collect PRM800K dataset and to perform our large-scale experiments.
However, our small-scale results in Section 4 suggest that large-scale models are
not necessary to observe benefits from process supervision. Second, we evaluate
on the MATH dataset, which is significantly more challenging than GSM8K.
Third, we collect a much larger quantity of process supervision data.
On the surface, the results from Uesato et al. (2022) may seem to conflict
with our claim that process supervision leads to better performance. However,
we believe the apparent conflict can be explained by the difference in the scale
of the supervision. The data scaling trend in Figure 4a suggests that a small
amount of process supervision and a large amount of outcome supervision do
in fact lead to similar performance, consistent with the results from Uesato
et al. (2022). The trend also shows that process supervision beats outcome
supervision when scaled up, even when judged based solely on outcomes. This
is consistent with our results in Section 3. We believe these results make a
strong case for using process supervision.
7.2 Synthetic Supervision
Similar to our work in Section 4, Gao et al. (2022) use a large reward model to
supervise the training of smaller models. They study the over-optimization that
occurs during RLHF, with experiments that require large quantities of human
preference data. To work around this challenge, they use a gold-standard reward
model to replace human feedback. Our use of a large-scale reward model to
supervise smaller reward models shares similarities with their approach.
7.3 Natural Language Reasoning
Several recent studies that have examined the reasoning ability of large language
models are implicitly relevant to our work. Lewkowycz et al. (2022) showed that
finetuning models on a large corpus of technical content led to significantly im-
proved performance on MATH. Wang et al. (2022) showed that self-consistency
12
leads to remarkably strong performance on many reasoning benchmarks, no-
tably without requiring any additional finetuning. Wei et al. (2022) and Nye
et al. (2021) demonstrate the importance of explicitly performing intermediate
reasoning steps via a chain of thought or a scratchpad in order to solve tasks
that require multi-step reasoning. Kojima et al. (2022) show that models are
able to perform this behavior zero-shot, conditioned only on a simple prompt.
8 Conclusion
We have shown that process supervision can be used to train much more reliable
reward models than outcome supervision in the domain of mathematical rea-
soning. We have also shown that active learning can be used to lower the cost of
human data collection by surfacing only the most valuable model completions
for human feedback. We release PRM800K, the full dataset of human feedback
used to train our state-of-the-art reward model, with the hope that removing
this significant barrier to entry will catalyze related research on the alignment of
large language models. We believe that process supervision is currently under-
explored, and we are excited for future work to more deeply investigate the
extent to which these methods generalize.
Acknowledgements
We thank Joshua Achiam, Mark Chen, Jonathan Gordon, Dan Hendrycks,
Lukasz Kaiser, Oleg Murk, Ben Sokolowsky, Francis Song, and Jonathan Uesato
for valuable feedback and thoughtful discussions; Giambattista Parascandolo
and Daniel Selsam for their contributions to the MathMix dataset; Jonathan
Ward for contributing to the data collection interface; Wojciech Zaremba for en-
couraging us to scale up data collection; Peter Hoeschele and Aris Kostantinidis
for supporting our data collection; the research acceleration and supercomput-
ing teams at OpenAI for providing infrastructure support; and the team at Scale
and the many data-labelers who created PRM800K.
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A MathMix
Similar to Lewkowycz et al. (2022) we construct a large-scale dataset of high-
quality math-relevant tokens for use in a lightweight pretraining stage, before
finetuning on comparably smaller datasets like MATH and PRM800K. This
dataset, which we call MathMix, has two main differences compared to the one
used to train Minerva. First, it is smaller and more aggressively filtered to high-
quality math problem-solving content, and second, it does not explicitly mix in
general language data.
Minerva was trained on 38.5B tokens of arXiv documents and webscrape
pages with LaTeX content, while MathMix consists of a smaller set of 1.5B
tokens containing individual math problems and their solutions, free-form text
discussing math problems and concepts, and synthetic data (Table 2). While
Minerva was pretrained on a dataset with 5% general natural language data,
we chose not to mix in any natural language data explicitly, primarily because
MathMix already contains plenty of natural language data.
Data type Token count Present in pretraining?
Math problems and solutions ∼275M No
Free-form math discussion text (1) ∼430M No
Free-form math discussion text (2) ∼450M Yes
Synthetic data (1) ∼30M No
Synthetic data (2) ∼100M Yes
Critiques grading data ∼500M No
Table 2: MathMix dataset components.
Note that when training smaller models, as in Section 4, we use a slightly
smaller variant of MathMix that excludes the critiques data and only consists of
1B tokens. For our large models experiments, we train on MathMix for roughly
3B tokens (2 epochs). For our small models experiments, we train for 6 epochs
(roughly 6.6B tokens).
We apply a set of decontamination checks on MathMix against the test
split of the MATH dataset, including stripping out LaTeX and searching for
matching n-grams, but we can make no strong guarantees on the efficacy of this
decontamination. As discussed in Section 6.3, we would not expect the relative
comparisons made throughout this work to be significantly impacted by test set
contamination.
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B PRM800K
We collected 1 ,085,590 step-level labels over 101 ,599 solution samples. We
present the whole unfiltered dataset as PRM800K. During training we discard
labels used for quality control, as well as any step-level labels for which the
labeler was unable to complete the task. The filtered dataset contains about
800,000 step-level labels over 75 ,000 solutions. The full PRM800K dataset is
available at https://github.com/openai/prm800k.
The data collection was split into two separate phases. In phase 1, we col-
lected labels for multiple alternative completions at each step of a solution. This
seeded our dataset but was cumbersome—for many steps the alternatives were
repetitive, and we found labelers spent a lot of time supervising long uninter-
esting solutions. As a result, the step-level labels we collected in this phase are
more repetitive than those collected later. In total, phase 1 represents about
5% of PRM800K, or about 40 ,000 step-level labels.
The majority of our labels were collected as part of phase 2, during which we
scaled up and streamlined the data collection process. Phase 2 data collection
is split into 10 generations. For each generation, we sample Nsolutions per
problem from the generator. We rank these solutions with our current best
PRM and surface the highest scoring wrong-answer solutions to our labelers.
We retrain this PRM between each generation using all the latest data. This
active learning strategy changes the balance of our data considerably. Though
we sometimes surfaced correct solutions (either by manually injecting correct
solutions or because of errors in our automatic grading), the vast majority of the
labels we collected in this phase are for incorrect solutions. Table 3 breaks down
the balance of correct/incorrect steps and solutions between the different phases
of data collection. Though we mostly collected labels on incorrect solutions, we
still collected many labels for correct individual steps. In fact, our small-scale
ablations in Section 4.2 suggest that this active learning strategy, which favors
labelling high-scoring wrong-answer solutions, improves performance despite the
resulting imbalance in the dataset.
phase 1 phase 2 combined
% end in correct solution 85 .1 13 .2 14 .2
% correct steps 58 .6 74 .1 73 .1
Table 3: Distribution of positive/negative steps/solutions.
Some of our phase 2 questions are intended for quality control. For a quality
control question, researchers mark which steps are reasonable to label as in-
correct. Then we assess that labelers are able to consistently mark those steps
as incorrect. Prior to starting on phase 2, we required all labelers to label 30
quality control questions. This served as a screening test, and we only admitted
labelers that agreed with our gold labels at least 75% of the time.
We then designated 10-20 problems per generation as additional quality
control questions, and we randomly served them to labelers as they worked
17
through the task. We used the results of this continuous quality control to
remove labelers whose quality slipped too far, as well as to prepare educational
material on common mistakes in order to improve labeler alignment with our
instructions.
C Evaluation
As we scaled up the project, we began having to collect labels on multiple
solutions for the same training problem. In order to avoid the risk of over-fitting
on the 7 ,500 MATH training problems, we expanded the training set to include
4,500 MATH test split problems. We therefore evaluate our models only on the
remaining 500 held-out problems. We selected these 500 test problems uniformly
at random. In Figure 5, we show that the distribution of difficulty levels and
subjects in this subset is representative of the MATH test set as a whole. The
specific test set we used can be found at https://github.com/openai/prm800k.
We leave it for future work to explore how many distinct training problems are
actually necessary, and how quickly our methods overfit to the training set.
Figure 5: Two histograms comparing the distribution of problem difficulty levels
and subjects in both the original MATH test set and in our 500 problem test
subset.
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D Labelling Instructions
Labelers were tasked to look at steps in a solution and label each one as posi-
tive,negative , orneutral . A step is considered neutral if it is appropriate in
context, reasonable, correct, and contains only computations that can be veri-
fied easily. A step is positive if it is neutral and also progresses towards the
solution. All other steps are considered negative . Labelers were not given ref-
erence solutions, but they were given the ground truth final answers. We chose
not to provide reference solutions to avoid biasing them towards one particular
path to the solution. We chose to provide ground truth final answers since this
information can sometimes help labelers resolve their own misunderstandings.
In phase 1, labelers were permitted to enter their own steps in the case that
all candidate steps were negative . Then the solution would progress from a
randomly selected positive step (or neutral if their were no positive ones).
This often resulted in trajectories that got stuck in endless sequences of neutral
steps that said reasonable things but made frustratingly slow progress towards a
solution or negative steps that needed constant human supervision. In phase 2,
we pre-generate whole solutions and end the task as soon as the first negative
step is encountered. The full instructions given to labelers can be found at
https://github.com/openai/prm800k/tree/main/prm800k/instructions.
E ORM Training Details
We train outcome-supervised reward models in the same manner as token-level
verifiers from Cobbe et al. (2021), with a few subtle differences to hyperparam-
eters. In particular, we only train for a single epoch on each dataset of model
samples and reward model labels, without dropout, and without jointly learn-
ing a language modeling objective. We find that performance is not sensitive to
most other hyperparameters, within a reasonable range.
To collect model samples, we simply sample uniformly from the generator at
a temperature of 1.0 without applying any rebalancing of positives or negatives.
At training time, the reward model makes predictions for every token in the
context. The target for each token in a solution is the same, based on whether
the solution is labelled correct or incorrect. At test time, we simply use the
score of the final token in the completion as the overall score of the solution.
We note that this setup is identical to the way token-level verifiers were trained
in Cobbe et al. (2021).
19
F PRM Details
F.1 Training
We train our PRMs by fine-tuning the MathMix model to predict the probability
of positive, negative, and neutral labels given a solution prefix ending in one of
our labeled steps. We sweep over hyperparameters using a dataset containing
the first ∼10% of PRM800K. Fine-tuning an LLM from its ordinary language
modeling task to a classification task like this is a large distribution shift, and
we found low learning rates were important to stable PRM training.
All of our PRMs are trained for 2 epochs. On smaller datasets (such as
in phase 1 and the first few generations of phase 2) this improves the final
performance over training for just 1 epoch. Additional epochs, up to some point,
don’t noticeably help or hurt performance. On larger datasets, the benefits of
2 epoch training diminishes, but we continue doing it for consistency.
F.2 Scoring
There are multiple ways of using the PRM to score solutions. In general, we
produce a single solution-level score by performing a reduction over step-level
scores, where the step-level score is the probability that the step’s label is pos-
itive. This involves two specific implementation decisions. First, when deter-
mining a step-level score, we either consider a neutral label to be positive or
negative. Second, when determining a solution-level score, we either use the
minimum or the product over step-level scores as a reduction.
We show results from all four scoring strategies in Table 4. The best per-
forming strategy is to take the product of step-level scores and to consider the
neutrals as positives, but the difference in performance between all strategies
is minor. Throughout the rest of this work, we consider neutral steps to be
positive, and we define the solution score to be the product of step-level scores.
Using the product instead of the minimum as the reduction does create a slight
bias against solutions with a larger number of steps.
product minimum
neutral = positive 78 .2% 77 .6%
neutral = negative 77 .4% 77 .8%
Table 4: Best-of-1860 test performance using the PRM with four different scor-
ing strategies.
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G Difficulty Breakdown
We show performance of our ORM and PRM on each quintile of the MATH
dataset. We determine quintiles based on the pass rate under the generator.
It is interesting to note that the performance gap is not only apparent on high
difficulty problems: it is in fact apparent across all difficulties. For the lowest
difficulty problems, we see that it is possible to find adversarial examples that
fool the ORM, since the ORM’s performance slightly decreases as the number of
samples increases. In contrast, the PRM remains highly robust over this same
set of samples.
We also see that increasing the number of samples has the largest positive
effect on the highest difficulty problems. This is to be expected, since a large
number of generator samples may be required to find a true and convincing
solution to a hard problem.
1011030.00.20.40.60.81.0% Problems Solved (Best-of-N)Quintile 1 (easiest)
PRM
ORM
101103Quintile 2
101103
N = number of solutions per problemQuintile 3
101103Quintile 4
101103Quintile 5 (hardest)
Figure 6: A breakdown of ORM vs PRM performance by problem difficulty.
21
H Synthetic Supervision Details
We can use PRM large to provide either outcome or process supervision for
smaller models. We determine the labels for individual steps based on the
step-level probabilities outputted by PRM large. To do this, we set an arbitrary
threshold: any step that PRM large assigns a negative label with greater than
20% probability is considered incorrect. We choose this threshold based on the
observation that PRM large is slightly miscalibrated in the direction of favoring
positive labels.
To provide process supervision for a solution, we directly return the step-
level labels (positive or negative) provided by PRM large, up until the first step
that is marked as negative. This mimics our true human data collection process.
To provide outcome supervision, we mark the solution as correct if and only if
PRM largeconsiders every step to be correct (using the same thresholding logic).
22
I PRM Visualizations
All examples shown come from the large-scale generator (GPT-4). We note
the pass-rate under the generator to give some sense of the difficulty of these
problems.
I.1 True Positives
These cherry-picked examples show the best-of-1860 solution from the generator
as ranked by the large-scale PRM.
Problem 1. Generator pass-rate: 0 .1%. This challenging trigonometry problem
requires applying several identities in a not-at-all obvious succession. Most
solution attempts fail, because it is hard to choose which identities are actually
helpful. Though successful solutions to this problem are rare, the reward model
correctly recognizes when a valid chain-of-thought has been found.
23
Problem 2. Generator pass-rate: 5 .8%. In step 7 and 8, the generator starts
performing guess-and-check. This is a common place the model might hallu-
cinate, by claiming a particular guess is successful when it isn’t. In this case,
the reward model verifies each step and determines that the chain-of-thought is
correct.
Problem 3. Generator pass-rate: 1 .7%. The generator successfully applies sev-
eral trigonometric identities to simplify the expression.
24
Problem 4. Generator pass-rate: 4 .5%. Here, the generator successfully per-
forms a complex series of polynomial factorizations. The use of the Sophie-
Germain identity in step 5 is an important step that could be considered in-
sightful.
I.2 True Negatives
Problem 5. Generator pass-rate: 4 .5%. The generator attempts to use the
difference of squares formula in step 12 on an expression that isn’t in fact a
difference of squares. The reward model catches this mistake.
25
Problem 6. Generator pass-rate: 93 .5%. In step 7, the generator makes an
incorrect attempt to simplify an expression. The reward model catches this
mistake.
Problem 7. Generator pass-rate: 48 .0%. In step 11, the generator makes a
simple calculation error. The reward model catches this mistake.
26
Problem 8. Generator pass-rate: 5 .8%. The justification in step 8 is strange,
but the reward model lets it slide. In step 9, though, the model incorrectly
factors the expression. The reward model catches this mistake.
I.3 False Positives
Problem 9. Generator pass-rate: 18 .5%. The generator makes a subtle counting
error in step 9. On the surface, it appears reasonable to claim that there are 5
ways to exchange the same colored ball since there are 5 colors. However, this
undercounts by a factor of 2, since Bob has 2 choices for which ball to return
to Alice. The reward model is fooled by this mistake.
27
Problem 10. Generator pass-rate: 17 .6%. In step 13, the generator attempts
to simplify the equation by combining like terms. It correctly moves and com-
bines the linear terms to the left-hand side, but then mistakenly leaves the
right-hand side untouched. The reward model is fooled by this mistake.
Problem 11. Generator pass-rate: 13 .4%. The generator attempts to per-
form long division, but in step 16, it forgets to include the leading zeros in the
repeating part of the decimal. The reward model is fooled by this mistake.
28
Problem 12. Generator pass-rate: 9 .1%. In step 4, the generator falsely
claims that the sequence repeats itself every 12 terms, when it’s in fact every
10 terms. This sort of counting mistake occasionally fools the reward model.
29

Bowen Baker
Teddy Lee
John Schulman
Greg Brockman
Kendra Rimbach
Hannah Wong
Thomas Degry