  State-of-the-art language models can exhibit impressive reasoning refinement
capabilities on math, science or coding tasks. However, recent work
demonstrates that even the best models struggle to identify \textit{when and
where to refine} without access to external feedback. Outcome-based Reward
Models (\textbf{ORMs}), trained to predict correctness of the final answer
indicating when to refine, offer one convenient solution for deciding when to
refine. Process Based Reward Models (\textbf{PRMs}), trained to predict
correctness of intermediate steps, can then be used to indicate where to
refine. But they are expensive to train, requiring extensive human annotations.
In this paper, we propose Stepwise ORMs (\textbf{SORMs}) which are trained,
only on synthetic data, to approximate the expected future reward of the
optimal policy or $V^{\star}$. More specifically, SORMs are trained to predict
the correctness of the final answer when sampling the current policy many times
(rather than only once as in the case of ORMs). Our experiments show that SORMs
can more accurately detect incorrect reasoning steps compared to ORMs, thus
improving downstream accuracy when doing refinements. We then train
\textit{global} refinement models, which take only the question and a draft
solution as input and predict a corrected solution, and \textit{local}
refinement models which also take as input a critique indicating the location
of the first reasoning error. We generate training data for both models
synthetically by reusing data used to train the SORM. We find combining global
and local refinements, using the ORM as a reranker, significantly outperforms
either one individually, as well as a best of three sample baseline. With this
strategy we can improve the accuracy of a LLaMA-2 13B model (already fine-tuned
with RL) on GSM8K from 53\% to 65\% when greedily sampled.
