To prevent signals of disparate numerical scales from dominating policy optimization, Evo-GDPO applies two successive normalization stages across the decoupled reward, latency, and cost channels:

1. Within-Group Normalization: For each metric $m \in \{\text{rew}, \text{lat}, \text{cost}\}$, the raw channel score $R_i^m$ is normalized across the group of $G$ sampled candidates: $A_i^m = \frac{R_i^m - \text{mean}(\{R_j^m\}_{j=1}^G)}{\text{std}(\{R_j^m\}_{j=1}^G) + \varepsilon_{\text{num}}}$

2. Batch-Level Aggregation and Normalization: The normalized metric advantages are combined via a weighted sum $A_i^\Sigma = w_{\text{rew}} A_i^{\text{rew}} + w_{\text{lat}} A_i^{\text{lat}} + w_{\text{cost}} A_i^{\text{cost}}$, where weights are non-negative, sum to 1, and enforce reward dominance ($w_{\text{rew}} > w_{\text{lat}} + w_{\text{cost}}$). The aggregated advantage is then stabilized across the training batch: $\widehat{A}_i^\Sigma = \frac{A_i^\Sigma - \text{mean}_{\text{batch}}(A^\Sigma)}{\text{std}_{\text{batch}}(A^\Sigma) + \varepsilon_{\text{num}}}$ where $\varepsilon_{\text{num}} > 0$ is a numerical stabilizer.