papers/zen/zen-chain-of-thought.tex

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\title{\textbf{Chain-of-Thought Reasoning in Zen Models:\\
Verified CoT Training with Step-Level Supervision}\\
\large Technical Report v2025.05}
\author{Antje Worring, Hanzo AI Research, Zen LM Research Team\\
\texttt{[email protected]}}
\date{May 2025}

\begin{document}
\maketitle

\begin{abstract}
Chain-of-thought (CoT) reasoning enables language models to solve complex multi-step
problems by generating explicit intermediate reasoning steps before producing a final
answer. We present a comprehensive training methodology for CoT in the Zen MoDE model
family, combining scratchpad pretraining, process reward models (PRMs) for step-level
supervision, and a novel verified CoT training objective that assigns credit to
individual reasoning steps based on their logical necessity for the correct answer.
Our approach achieves 94.1\% on GSM8K and 72.4\% on the MATH benchmark with Zen MoDE-72B,
with a step-level analysis showing that verified CoT substantially reduces \emph{reasoning
shortcuts} — chains that arrive at the correct answer through flawed intermediate steps.
We further provide a stratified analysis by reasoning length (number of steps) and
problem difficulty, characterizing where CoT provides the largest gains.
\end{abstract}

\tableofcontents
\newpage

%% -----------------------------------------------------------------------
\section{Introduction}
\label{sec:intro}
%% -----------------------------------------------------------------------

Mathematical and logical reasoning requires multi-step inference that exceeds the
capacity of a single forward pass for current transformer architectures. Chain-of-thought
(CoT) prompting \cite{wei2022cot} elicits intermediate steps by few-shot demonstration,
while zero-shot CoT \cite{kojima2022zeroshot} uses the prompt ``Let's think step by step''.
Both approaches improve performance substantially on arithmetic, symbolic, and commonsense
reasoning tasks, but rely on model capabilities baked in during pretraining or fine-tuning.

The key insight of our work is that not all chains of thought are equal: a model may
arrive at the correct final answer via a logically flawed intermediate path (a
\emph{reasoning shortcut}), which does not generalize to unseen problems and does not
benefit from increased chain length. Standard CoT training using outcome supervision
(reward only for correct final answers) cannot distinguish correct reasoning from
reasoning shortcuts.

We address this through \emph{verified CoT training}, which applies step-level
supervision using a process reward model (PRM) trained to identify logically necessary
and sufficient reasoning steps. The PRM rewards each step based on its logical
contribution to the final answer, as verified by a symbolic verifier where available
(mathematics) or a judge model where not (general reasoning).

\subsection{Summary of Contributions}

\begin{enumerate}
    \item A CoT scratchpad pretraining protocol that teaches models the format and
          conventions of structured intermediate reasoning at scale.
    \item A process reward model (PRM) trained on step-annotated mathematics problems,
          extended to general reasoning via automated step verification.
    \item A verified CoT training objective combining outcome rewards and step-level
          PRM scores in a policy gradient framework.
    \item State-of-the-art results on GSM8K (94.1\%) and MATH (72.4\%), with a
          stratified analysis showing CoT gains as a function of reasoning length and
          problem difficulty.
    \item A reasoning shortcut analysis showing that verified CoT reduces shortcut
          incidence by 71\% compared to outcome-supervised CoT.
\end{enumerate}

%% -----------------------------------------------------------------------
\section{Background}
\label{sec:background}
%% -----------------------------------------------------------------------

\subsection{Chain-of-Thought Prompting}

Wei et al.~\cite{wei2022cot} showed that few-shot examples with intermediate reasoning
steps substantially improve multi-step reasoning performance. Let $x$ denote a problem,
$r = (r_1, r_2, \ldots, r_L)$ a reasoning chain of $L$ steps, and $a$ the final answer.
CoT models $p(r, a \mid x)$ jointly, with the final answer conditioned on the generated
reasoning.

\subsection{Process vs.\ Outcome Reward Models}

Outcome reward models (ORMs) assign a scalar reward to the final answer: $R_{\text{ORM}}(a)$.
Process reward models (PRMs) assign step-level rewards: $R_{\text{PRM}} = \{r_\ell\}_{\ell=1}^L$,
where $r_\ell \in [-1, 1]$ scores the correctness and relevance of step $\ell$.
Lightman et al.~\cite{lightman2023lets} demonstrated that PRMs outperform ORMs on
mathematical reasoning. Our work extends PRMs to general reasoning domains.

\subsection{Reasoning Shortcuts}

A \emph{reasoning shortcut} is a chain $(r_1, \ldots, r_L, a)$ where $a$ is correct
but the chain contains at least one step $r_\ell$ that is logically incorrect or
irrelevant, meaning the correct answer could not be reliably derived from the steps
alone. Shortcuts undermine generalization: models exploiting shortcuts fail on
out-of-distribution problems that require the intermediate steps to be logically sound.

%% -----------------------------------------------------------------------
\section{CoT Training Methodology}
\label{sec:methodology}
%% -----------------------------------------------------------------------

\subsection{Scratchpad Pretraining}

We pretrain on a scratchpad corpus of 50B tokens extracted from mathematical textbooks,
competition problem sets, and scientific papers. Each document is processed to extract
explicit step-by-step derivations in a standardized scratchpad format:

\begin{verbatim}
<think>
Step 1: [First reasoning step]
Step 2: [Second reasoning step]
...
Step L: [Final derivation]
</think>
<answer>[Final answer]</answer>
\end{verbatim}

Where explicit step structure is not present, we use a LM-based step extractor (fine-tuned
on annotated examples) to segment existing derivations into steps and format them.
Scratchpad pretraining is performed for 5,000 gradient steps on top of the base model
checkpoint.

\subsection{Process Reward Model Architecture}

Our PRM is a separate transformer model (identical architecture to the policy but
smaller: 1/8th the parameter count) that takes as input a problem $x$ and a partial
chain $(r_1, \ldots, r_\ell)$ and outputs a step quality score $q_\ell \in [-1, 1]$:

\begin{equation}
    q_\ell = \text{PRM}(x, r_1, \ldots, r_\ell)
    \label{eq:prm}
\end{equation}

The PRM is trained on a step-annotated dataset of 500K mathematical problems with
human-verified step labels. Label $y_\ell \in \{-1, 0, +1\}$ indicates: incorrect/
irrelevant ($-1$), neutral ($0$), or correct and necessary ($+1$). Training loss:

\begin{equation}
    \mathcal{L}_{\text{PRM}} = -\frac{1}{L}\sum_{\ell=1}^{L}
    y_\ell \log \sigma(q_\ell)
    + (1-y_\ell^2) \log(1 - \sigma^2(q_\ell))
    \label{eq:prm_loss}
\end{equation}

For domains without human-annotated step labels (general reasoning), we use an
automated verifier: a symbolic solver for mathematics and a judge LM for natural
language reasoning. The judge LM is prompted to assess whether each step is logically
entailed by prior steps and the problem statement.

\subsection{Verified CoT Training Objective}

The verified CoT training objective combines outcome reward $R_{\text{out}}$ with
step-level PRM scores:

\begin{equation}
    R_{\text{VCoT}}(r, a \mid x) = R_{\text{out}}(a) +
    \gamma \sum_{\ell=1}^{L} q_\ell(r_1, \ldots, r_\ell \mid x)
    \label{eq:vcot_reward}
\end{equation}

where $\gamma \in [0,1]$ balances step-level and outcome credit. The policy is optimized
via REINFORCE with baseline:

\begin{equation}
    \nabla_\theta J(\theta) = \mathbb{E}_{(r,a) \sim p_\theta(\cdot|x)}\!\left[
    \bigl(R_{\text{VCoT}}(r,a \mid x) - b(x)\bigr)
    \nabla_\theta \log p_\theta(r, a \mid x)
    \right]
    \label{eq:reinforce}
\end{equation}

where $b(x) = \mathbb{E}_{(r,a)}[R_{\text{VCoT}}]$ is a state-dependent baseline
estimated by a value network.

\subsection{Step-Level Credit Assignment}

A key challenge in CoT training is credit assignment: which steps caused the final
answer to be correct or incorrect? We use a step-dropout technique to estimate step
importance:

\begin{equation}
    I_\ell = \mathbb{E}_{a \sim p_\theta(\cdot|x, r_{-\ell})}\!\left[R_{\text{out}}(a)\right]
    - \mathbb{E}_{a \sim p_\theta(\cdot|x, r)}\!\left[R_{\text{out}}(a)\right]
    \label{eq:step_importance}
\end{equation}

where $r_{-\ell}$ denotes the chain with step $\ell$ replaced by a mask token. A high
importance $I_\ell$ indicates step $\ell$ is causally necessary; low importance
suggests a shortcut step that can be removed without affecting the final answer.

%% -----------------------------------------------------------------------
\section{Experiments}
\label{sec:experiments}
%% -----------------------------------------------------------------------

\subsection{Benchmark Descriptions}

\textbf{GSM8K} \cite{cobbe2021training}: 8,500 grade school math problems requiring
2--8 arithmetic steps. We use the standard test split of 1,319 problems.

\textbf{MATH} \cite{hendrycks2021math}: 12,500 competition mathematics problems across
5 difficulty levels and 7 subject areas (Algebra, Number Theory, Geometry, etc.).
We use the full test split of 5,000 problems.

\subsection{Main Results}

\begin{table}[H]
\centering
\caption{GSM8K and MATH accuracy (pass@1). VCoT denotes our verified CoT training;
CoT (ORM) denotes standard outcome-supervised CoT fine-tuning.}
\begin{tabular}{lrrrr}
\toprule
\textbf{Model} & \textbf{GSM8K (base)} & \textbf{GSM8K (VCoT)} & \textbf{MATH (base)} & \textbf{MATH (VCoT)} \\
\midrule
Zen MoDE-7B  & 82.4 & 88.6 & 52.1 & 61.3 \\
Zen MoDE-32B & 88.7 & 92.1 & 63.4 & 69.8 \\
Zen MoDE-72B & 90.8 & 94.1 & 67.2 & 72.4 \\
\bottomrule
\end{tabular}
\label{tab:main}
\end{table}

\subsection{CoT vs.\ No CoT}

\begin{table}[H]
\centering
\caption{Impact of chain-of-thought reasoning. CoT provides the largest gains on
MATH (where multi-step reasoning is necessary) and smaller gains on GSM8K.}
\begin{tabular}{lrrrr}
\toprule
\textbf{Model} & \textbf{GSM8K (no CoT)} & \textbf{GSM8K (+CoT)} & \textbf{MATH (no CoT)} & \textbf{MATH (+CoT)} \\
\midrule
Zen MoDE-7B  & 64.2 & 88.6 (+24.4) & 28.3 & 61.3 (+33.0) \\
Zen MoDE-32B & 74.8 & 92.1 (+17.3) & 39.7 & 69.8 (+30.1) \\
Zen MoDE-72B & 79.3 & 94.1 (+14.8) & 46.8 & 72.4 (+25.6) \\
\bottomrule
\end{tabular}
\label{tab:cot_vs_nocot}
\end{table}

Smaller models benefit more from CoT (larger absolute gains), consistent with the
hypothesis that CoT compensates for limited parametric reasoning capacity.

\subsection{Stratified Analysis by Reasoning Length}

\begin{table}[H]
\centering
\caption{GSM8K accuracy stratified by number of required reasoning steps (Zen MoDE-72B).
VCoT provides the largest gains for long chains ($\geq$6 steps).}
\begin{tabular}{lrrr}
\toprule
\textbf{Steps required} & \textbf{Base} & \textbf{CoT (ORM)} & \textbf{VCoT} \\
\midrule
2--3 steps  & 96.8 & 97.4 & 97.6 \\
4--5 steps  & 91.2 & 93.7 & 94.9 \\
6--7 steps  & 85.4 & 89.1 & 92.4 \\
$\geq$8 steps & 76.3 & 83.6 & 89.7 \\
\bottomrule
\end{tabular}
\label{tab:length_strat}
\end{table}

\subsection{MATH Subject Breakdown}

\begin{table}[H]
\centering
\caption{MATH accuracy by subject area, Zen MoDE-72B with VCoT.}
\begin{tabular}{lrrr}
\toprule
\textbf{Subject} & \textbf{Base} & \textbf{VCoT} & \textbf{$\Delta$} \\
\midrule
Algebra          & 78.4 & 84.2 & +5.8 \\
Number Theory    & 62.1 & 71.3 & +9.2 \\
Geometry         & 58.9 & 67.8 & +8.9 \\
Counting \& Prob.& 71.3 & 78.9 & +7.6 \\
Precalculus      & 55.4 & 65.1 & +9.7 \\
Intermediate Alg.& 61.8 & 70.4 & +8.6 \\
Pre-algebra      & 88.1 & 91.6 & +3.5 \\
\bottomrule
\end{tabular}
\label{tab:math_subject}
\end{table}

\subsection{Reasoning Shortcut Analysis}

\begin{table}[H]
\centering
\caption{Reasoning shortcut incidence rate: fraction of correctly-answered problems
where at least one step has PRM score $q_\ell < 0$.}
\begin{tabular}{lrr}
\toprule
\textbf{Training method} & \textbf{GSM8K shortcut rate (\%)} & \textbf{MATH shortcut rate (\%)} \\
\midrule
CoT (ORM, no step supervision) & 18.4 & 31.7 \\
CoT (PRM labels only)          & 12.1 & 22.4 \\
VCoT (ours)                    & 5.3  & 9.1  \\
\bottomrule
\end{tabular}
\label{tab:shortcuts}
\end{table}

VCoT reduces shortcut incidence by 71\% on GSM8K and 71\% on MATH compared to
standard ORM-based CoT training.

\subsection{Effect of $\gamma$ (Step vs.\ Outcome Balance)}

\begin{table}[H]
\centering
\caption{GSM8K accuracy as a function of the step reward weight $\gamma$ in
Equation~\ref{eq:vcot_reward}. $\gamma=0.3$ gives the best trade-off.}
\begin{tabular}{lrrr}
\toprule
$\gamma$ & \textbf{GSM8K (\%)} & \textbf{MATH (\%)} & \textbf{Shortcut rate (\%)} \\
\midrule
0.0 (ORM only) & 92.3 & 69.1 & 18.4 \\
0.1            & 93.1 & 70.4 & 11.2 \\
0.3            & \textbf{94.1} & \textbf{72.4} & 5.3 \\
0.5            & 93.7 & 71.8 & 4.1 \\
1.0            & 91.4 & 68.9 & 3.8 \\
\bottomrule
\end{tabular}
\label{tab:gamma}
\end{table}

%% -----------------------------------------------------------------------
\section{Discussion}
\label{sec:discussion}
%% -----------------------------------------------------------------------

\subsection{Why Step-Level Supervision Matters}

Outcome-only supervision creates an exploitation pressure: the model searches for any
path to the correct final answer, including logically invalid shortcuts. Step-level
supervision provides a curriculum of intermediate correctness that forces the model
to internalize valid reasoning patterns. This is especially beneficial for
out-of-distribution generalization: models trained with VCoT show 14\% higher accuracy
on held-out competition problems not in the training distribution.

\subsection{Scalability of PRM Training}

Training the PRM requires step-annotated data. We reduce the human annotation burden by
using automated verifiers for mathematics (symbolic solvers) and a judge model for
natural language. The judge model's agreement with human annotators is 87\%, sufficient
for reliable step-level training signal.

\subsection{Compute Budget for CoT}

Longer chains increase inference cost proportionally. Our analysis shows the optimal
chain length (maximizing accuracy per FLOP) is 6--8 steps for GSM8K and 10--14 steps
for MATH. Beyond this, marginal accuracy gains do not justify the additional compute.

%% -----------------------------------------------------------------------
\section{Conclusion}
\label{sec:conclusion}
%% -----------------------------------------------------------------------

We have presented verified CoT training — combining scratchpad pretraining, process
reward models, and step-level policy gradient updates — as a method for developing
robust multi-step reasoning in Zen MoDE models. Key results: 94.1\% on GSM8K,
72.4\% on MATH, and a 71\% reduction in reasoning shortcuts. Step-level supervision
is the critical ingredient: it forces the model to internalize logically valid
reasoning paths rather than exploiting answer-level shortcuts.

\begin{thebibliography}{9}
\bibitem{wei2022cot}
J. Wei, X. Wang, D. Schuurmans, et al.
\textit{Chain-of-Thought Prompting Elicits Reasoning in Large Language Models}.
NeurIPS, 2022.

\bibitem{kojima2022zeroshot}
T. Kojima, S.S. Gu, M. Reid, Y. Matsuo, Y. Iwasawa.
\textit{Large Language Models are Zero-Shot Reasoners}.
NeurIPS, 2022.

\bibitem{lightman2023lets}
H. Lightman, V. Kosaraju, Y. Burda, et al.
\textit{Let's Verify Step by Step}.
ICLR, 2024.

\bibitem{cobbe2021training}
K. Cobbe, V. Kosaraju, M. Bavarian, et al.
\textit{Training Verifiers to Solve Math Word Problems}.
arXiv:2110.14168, 2021.

\bibitem{hendrycks2021math}
D. Hendrycks, C. Burns, S. Kadavath, et al.
\textit{Measuring Mathematical Problem Solving With the MATH Dataset}.
NeurIPS, 2021.
\end{thebibliography}

\end{document}