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@@ -313,6 +313,49 @@ Chinese--English instruction dataset decreases from $\sim$2.5 to
 $\sim$1.6 over 1{,}000 steps, with the gradient norm converging to a
 stable range after the warmup phase.
 
+\subsection{Direct Preference Optimization}
+\label{sec:dpo}
+
+Following supervised fine-tuning, we align the model with pairwise
+human preferences via Direct Preference Optimization
+(DPO)~\cite{rafailov2023dpo}.  DPO avoids explicit reward-model training
+by optimizing the policy $\pi$ directly against a frozen reference
+policy $\pi_{\text{ref}}$ (the SFT checkpoint).  For a prompt $x$ with
+preferred response $y_w$ and dispreferred response $y_l$, the loss is:
+\begin{equation}
+\mathcal{L}_{\text{DPO}} = -\log \sigma\!\left(
+\beta \Bigl[
+\log\tfrac{\pi(y_w\mid x)}{\pi_{\text{ref}}(y_w\mid x)}
+-\log\tfrac{\pi(y_l\mid x)}{\pi_{\text{ref}}(y_l\mid x)}
+\Bigr]\right),
+\end{equation}
+where $\beta$ controls the KL-divergence penalty against the reference.
+We use $\beta=0.1$, a batch of 64 preference pairs per step, and the
+same hybrid Muon/AdamW optimizer with cosine scheduling.
+
+\begin{figure}[H]
+\centering
+\includegraphics[width=0.95\linewidth]{data/dpo_metrics.png}
+\caption{DPO training metrics over $\sim$1{,}500 alignment steps:
+preference loss (raw and 100-step moving average), learning rate
+schedule, and gradient norm.}
+\label{fig:dpo_metrics}
+\end{figure}
+
+Figure~\ref{fig:dpo_metrics} summarises the DPO training dynamics.
+The raw preference loss (left panel) starts near $0.69$ and is visibly
+noisy, a hallmark of pairwise preference sampling.  The 100-step moving
+average reveals a steady downward trend that reaches a minimum of
+$\sim$0.51 near step 1{,}200, after which it gently rebounds to
+$\sim$0.52 and oscillates, suggesting the policy has converged to a
+stable preference boundary rather than over-optimising away from the
+reference distribution.  The learning-rate schedule (centre panel) peaks
+at $5\times10^{-6}$ after a short linear warmup and then follows cosine
+decay to a floor of $\sim$0.5\,$\times\,$10$^{-6}$.  Gradient norms (right panel) start
+near 50 with occasional spikes above 55, then gradually decline and
+stabilise in the 35--40 range after step 400, indicating consistent
+gradient magnitudes throughout alignment.
+
 % ======================================================================
 \section{Numerical Stability via Residual Scaling}
 \label{sec:num-stability}
@@ -808,6 +851,11 @@ A.~Radford, J.~Wu, R.~Child, D.~Luan, D.~Amodei, I.~Sutskever.
 Language models are unsupervised multitask learners.
 \textit{OpenAI Blog}, 2019.
 
+\bibitem{rafailov2023dpo}
+R.~Rafailov, A.~Sharma, E.~Mitchell, C.~D.~Manning, S.~Ermon, C.~Finn.
+Direct Preference Optimization: Your language model is secretly a reward model.
+\textit{NeurIPS}, 2023.
+
 \bibitem{shazeer2020glu}
 N.~Shazeer. GLU variants improve Transformer.
 \textit{arXiv:2002.05202}, 2020.