Let $p_1,\ldots,p_N,q_1,\ldots,q_N\in\mathbb{R}^d$ be fixed parameters, $X\in\mathbb{R}^d$ be a vector of $d$ independent normal random variables.
How can we show the existance of $L$ and some matrix norm $||\cdot||$ such that
$$ \mathbb{E}_X \sum_{i=1}^N\left|\frac{\exp(q_i^TX)}{\sum_{j=1}^N\exp(q_j^TX)} - \frac{\exp(p_i^TX)}{\sum_{j=1}^N\exp(p_j^TX)} \right| \le L \cdot ||\{q_1,\ldots,q_N\}-\{p_1,\ldots,p_N\}||. $$