​
近期在学习 MagicAgent 这篇论文的时候，在第四章接提到：模型在合成数据的不同子集上进行训练。为了确保数据的多样性，该框架采用了 NovelSum 指标，根据候选样本 (candidate samples)的邻近权重和密度因子计算其新颖度得分。

原文链接：https://arxiv.org/abs/2602.19000

原文相关字段：

We adopt the Qwen3 series as the base model and further train it using our synthetic dataset, including the Hierarchical Task Decomposition, Tool-Augmented Planning, Multi-Constraint Scheduling, Procedural Logic Orchestration, and Long-Horizon Tool Execution tasks. To ensure comprehensive coverage and diversity across domains, we employ the NovelSum metric (Yang et al., 2025b) to construct a representative sub-training set $D_{sample}$, thereby facilitating the effective fusion of heterogeneous planning data. Specifically, the novelty of a candidate sample x relative to the currently selected training set $D_{sample}$ is defined as follows:
$$\begin{array}{c}v(x)=\sum _{x_j\in {\mathcal{D}}_{\text{sample}}}w(x,x_j{)}^{{\kappa }_1}\cdot \sigma (x_j{)}^{{\kappa }_2}\cdot d(x,x_j),\end{array}\text{\hspace{1em}(1)}$$
where the proximity weight $w(x,x_j)=\frac{1}{\pi (j)}$ assigns higher importance to closer neighbors, with $\pi (j)$indicating the rank of $x_j$ when samples are ordered by increasing distance to $x$. The density factor is defined as $\sigma (x_j)=\frac{1}{{\sum }_{k=1}^{K}d(x_j,N_k(x_j))}$ , $N_k(x_j)$denotes the k-th nearest neighbor of $x$, and $d(\cdot ,\cdot )$ denotes the distance between sample embeddings. The hyper-parameters κ1 and κ2 control the degree of the proximity weight and density factor, respectively. At each iteration, the sample $x_{high}$with the highest novelty score is selected with $x_{\text{high}}={\arg \max }_{x}v(x)$, and added to the training set as ${\mathcal{D}}_{\text{sample}}\leftarrow {\mathcal{D}}_{\text{sample}}\cup x_{\text{high}}$. This procedure is repeated, starting from $D_{sample}=\varnothing$, until the predefined data budget is met.

The loss function formulation for the optimization objective of the SFT stage is defined as the standard token-level cross-entropy loss:

$$\begin{array}{c}{L}_{\text{sft}}=-\frac{1}{N}\sum _{i=1}^N\sum _{j=1}^V{y}_{ij}\log (p_{ij}),\end{array}\text{\hspace{1em}(2)}$$

where $N$ is the sequence length, $V$ is the vocabulary size, $y_{ij}$ is the ground-truth indicator for token $i$, and $p_{ij}$ is the predicted probability for class $j$ at token $i$.

​
我先想通过一个实例去尝试计算这个公式来选 candidate 数据到 D_sample 中，但是我发现这个公式不知道怎么带入计算。我现在假设我有$x_1,x_2,x_3,x_4,x_5$这五个数据。我给出来这五个数据之间的 d 值，见下表：

在初始状态中，我分别设超参数 $k1=1,k2=1$.此时$D_{sample}=\varnothing$，采用非严谨计算方式，我这里直接将原本需要随机选择的第一个样本定为$x_1$，也就是Dsample={x1}D_{sample} = \{x_1\}Dsample​={x1​}。
接下来第一轮，我们需要通过上面提到的公式（1）来计算$v(x)$的值：
根据相关定义，我们计算
$v(x_2)=w(x_2,x_1)\ast \sigma (x_1)\ast d(x_2,x_1)=1\ast 0.2\ast \sigma (x_1)=\frac{1}{{\sum }_{k=1}^{K}d(x_1,N_k(x_1))}\ast 0.2$
但是到了这一步我发现我不知道怎么计算分母中${\sum }_{k=1}^{K}d(x_1,N_k(x_1))$怎么去计算。

希望各位大佬高抬贵手，帮小弟指导一下。