machine learning - Derivative of log-likelihood function in softmax regression

Question

I'm trying to find the derivative of the log-likelihood function in softmax regression. I have (with $\Theta$ being the parameters, and $x^{(i)}$ being the $i$th training example, and $s_j$ representing the softmax function),

$$\ell(\Theta) = \sum_{i=1}^m \sum_{j=1}^k \log \left( \frac{e^{\Theta_j^T x^{(i)}}}{\sum_{l=1}^k e^{\Theta_l^T x^{(i)}}} \right)^{I\{y^{(i)}=j\}}$$ I got the derivative of the softmax function itself as
$$\frac{\partial}{\partial \Theta_p} \left( \frac{e^{\Theta^T_j x^{(i)}}}{\sum_{l=1}^k e^{\Theta^T_l x^{(i)}}} \right)=s_j(\delta_{pj}-s_p)x^{(i)}$$
On using this to find the derivative of the log-likelihood, I get $$\begin{aligned}\frac{\partial}{\partial \Theta_p}\ell(\Theta) &= \sum_{i=1}^m \sum_{j=1}^k I\{y^{(i)}=j\} \cdot \frac{s_j(\delta_{pj}-s_p)x^{(i)}}{s_j} \\ &= \sum_{i=1}^m \sum_{j=1}^k I\{y^{(i)}=j\} \cdot (\delta_{pj}-s_p)x^{(i)} \\ &= \sum_{i=1}^m \sum_{j=1}^k I\{y^{(i)}=p\}x^{(i)}-I\{y^{(i)}=j\}s_p x^{(i)} \end{aligned} $$
I'm not sure where to go from here. From what I've seen online, I shouldn't have the second summation at all, and the second term should just be $s_p$.

I'm sure I'm just missing a step or two, but I'd love if someone could help me move ahead.

Answer 1

You are correct up to the second line of your working in the last part, and then you make an error by dropping the requirement that $j=p$ (which means you retain an additional sum that shouldn't be there). Continuing from your last correct step, you should have:

$$\begin{aligned} \frac{\partial \ell}{\partial \Theta_p}(\Theta) &= \sum_{i=1}^m \sum_{j=1}^k \mathbb{I}(y^{(i)}=j) (\delta_{pj}-s_p) x^{(i)} \\[6pt] &= \sum_{i=1}^m x^{(i)} \sum_{j=1}^k \mathbb{I}(y^{(i)}=j) (\delta_{pj}-s_p) \\[6pt] &= \sum_{i=1}^m x^{(i)} \bigg[ \sum_{j=1}^k \mathbb{I}(y^{(i)}=j) \mathbb{I}(p=j) - s_p \sum_{j=1}^k \mathbb{I}(y^{(i)}=j) \Bigg] \\[6pt] &= \sum_{i=1}^m x^{(i)} \bigg[ \mathbb{I}(y^{(i)}=p) - s_p \Bigg] \\[6pt] &= \sum_{i=1}^m x^{(i)} \mathbb{I}(y^{(i)}=p) - s_p \sum_{i=1}^m x^{(i)}. \\[6pt] \end{aligned} $$

(The penultimate step follows from the fact that $\sum_{j=1}^k \mathbb{I}(y^{(i)}=j) = 1$ for all $i = 1, ..., m$.)