Fix LaTeX rendering in section 2 of thinking mid-training blog

projects/thinking_midtraining/README.md

@@ -58,7 +58,11 @@ $\tilde{\mathcal{D}} = \{\tilde{c}^1, \tilde{c}^2, \ldots, \tilde{c}^N\}$.
 
 ### 2) Thinking SFT Mid-training
 
-We perform supervised fine-tuning (SFT) mid-training on half of the augmented corpus, which we call $\tilde{\mathcal{D}}_{\text{SFT}}$, using standard next-token prediction. Given a base model $\mathcal{M}_{\text{base}}$ parameterized by $\theta$, we optimize the following objective: $\mathcal{L}_{\text{SFT}}(\theta) = -\mathbb{E}_{\tilde{c}^i \sim \tilde{\mathcal{D}}} \left[ \sum_{j=1}^{|\tilde{c}^i|} \log P_\theta(\tilde{c}^i_j \mid \tilde{c}^i_{<j}) \right]$
+We perform supervised fine-tuning (SFT) mid-training on half of the augmented corpus, which we call $\tilde{\mathcal{D}}_{\text{SFT}}$, using standard next-token prediction.
+
+Given a base model $\mathcal{M}_{\text{base}}$ parameterized by $\theta$, we optimize the following objective:
+
+$$\mathcal{L}_{\text{SFT}}(\theta) = -\mathbb{E}_{\tilde{c}^i \sim \tilde{\mathcal{D}}} \left[ \sum_{j=1}^{|\tilde{c}^i|} \log P_\theta(\tilde{c}^i_j \mid \tilde{c}^i_{<j}) \right]$$
 
 where $\tilde{c}^i_j$ denotes the $j$-th token in the augmented chunk $\tilde{c}^i$, and $\tilde{c}^i_{<j}$ represents all preceding tokens. Importantly, the loss is computed over the entire augmented sequence, including both the original content tokens $x_j$ and the generated thought tokens $\tau_j$. This allows the model to learn to produce intermediate reasoning steps alongside the original content.