Free Process Rewards without Process Labels
Authors: name: Lifan Yuan, name: Wendi Li, name: Huayu Chen, name: Ganqu Cui, name: Ning Ding, name: Kaiyan Zhang, name: Bowen Zhou, name: Zhiyuan Liu, name: Hao Peng

Different from its counterpart outcome reward models (ORMs), which evaluate
the entire responses, a process reward model (PRM) scores a reasoning
trajectory step by step, providing denser and more fine grained rewards.
However, training a PRM requires labels annotated at every intermediate step,
presenting significant challenges for both manual and automatic data
collection. This paper aims to address this challenge. Both theoretically and
empirically, we show that an implicit PRM can be obtained at no
additional cost, by simply training an ORM on the cheaper response-level
labels. The only assumption is to parameterize the outcome reward as the
log-likelihood ratios of the policy and reference models, which can be
optimized regardless of the specific choice of loss objectives. In experiments,
we instantiate our implicit PRMs with various objectives and evaluate their
performance on MATH. We show that our implicit PRM outperforms a strong
MCTS-based baseline \'a la Math-Shepherd using less than 1/38 of the
training data. Its performance can be further improved with majority voting. We
further find that scaling up instructions and responses benefits our implicit
PRM, and the latter brings a larger gain. Particularly, we find that our
implicit PRM, when instantiated with the cross-entropy (CE) loss, is more
data-efficient and can keep improving generation models even when trained with
only one response per instruction, the setup that suffers from extreme data
scarcity and imbalance. Further, instructions should be relevant to downstream
tasks while the diversity of responses does not bring gains. Surprisingly,
training on extra Math-Shepherd step labels brings no further improvements to
our implicit PRM trained on only outcome data. We hope that our work will
encourage a rethinking of PRM training approaches and contribute to making
training PRMs more accessible.