I'm reading the Stanford course about Convolutional Neural Network and I don't understand how he backpropagates a 2 neural network. Actually, the thing I'm trying to understand is here: http://cs231n.github.io/neural-networks-case-study/

We know that:

scores = np.dot(X, W) + b


And to backpropagate he computes:

dW = np.dot(X.T, dscores)
db = np.sum(dscores, axis=0, keepdims=True)


So mathematically, we can write ($F$ being scores) $$F = XW + b$$

and when he backpropagates he gets: $${dW} = X^\intercal {dF}$$ $${db} = \begin{bmatrix} \sum\limits_{i=1} {dF}_{1i} \\ \sum\limits_{i=1} {dF}_{2i} \\ \vdots \\ \sum\limits_{i=1} {dF}_{ni} \end{bmatrix}$$

I'm trying to do the math but I don't understand how to achieve those derivatives rigorously. I hope some of you will help me to understand how it works :s

Consider the loss function for the network evaluated at the $i$th observation

\begin{align} \mathcal{L}_i &= - \log\left(p_{y_i}\right), \end{align}

where $p_{y_i}$ is the vector of normalised probabilities

\begin{align} p_{y_i} &= \frac{e^{f_{y_i}}}{\sum_{j=1}^K e^{f_j}}, \end{align}

$f_j = \boldsymbol{w_j}^\top \boldsymbol{x_i} + b_j$ is the score of the $i$th observation corresponding to the $j$th class, $\boldsymbol{x_i}$ is the $D$-dimensional column vector corresponding to the $i$th observation, and $\boldsymbol{w_j}$ is the column vector corresponding to the weights for the $j$th class, for $i=1,\cdots,N$ and $j=1,\cdots,K$.

Given the upstream derivative of the loss with respect to the scores

\begin{align} \frac{\partial\mathcal{L}_i}{\partial f_k} &= p_k - \mathbb{1}\left(y_i=k\right) \end{align}

we're interested in deriving the derivative of the loss with respect to the weights and biases. Starting with the weights, we can use the chain rule to get to the desired derivative via the derivatives of the loss with respect to the scores

\begin{align} \frac{\partial\mathcal{L}_i}{\partial \boldsymbol{w_j}} &= \sum_{k=1}^K \frac{\partial\mathcal{L}_i}{\partial f_k}\times\frac{\partial f_k}{\partial \boldsymbol{w_j}}\\ &= \frac{\partial\mathcal{L}_i}{\partial f_j}\times\frac{\partial f_j}{\partial \boldsymbol{w_j}}\\ &= \frac{\partial\mathcal{L}_i}{\partial f_j}\boldsymbol{x_i},\\ \end{align}

which follows from the derivative $\frac{\partial f_k}{\partial \boldsymbol{w_j}} = \boldsymbol{x_j}\mathbb{1}\left(j=k\right)$. The $j$th score is the only score where the vector $\boldsymbol{w_j}$ arises, and so the derivatives of the other scores with respect to $\boldsymbol{w_j}$ are all zero. The total loss $\mathcal{L}$ is obtained as the summation of the losses across all $N$ observations, and so

\begin{align*} \frac{\partial\mathcal{L}}{\partial \boldsymbol{w_j}} &= \sum_{i=1}^N \frac{\partial\mathcal{L}_i}{\partial f_j}\boldsymbol{x_i}. \end{align*}

This matrix of the derivatives of the loss with respect to the weights is simply obtained as the matrix product of the observation matrix and the matrix of the derivatives of the loss with respect to the scores \begin{align*} \begin{bmatrix} \sum_{i=1}^N \frac{\partial\mathcal{L}_i}{\partial f_1}\boldsymbol{x_i} & \cdots & \sum_{i=1}^N \frac{\partial\mathcal{L}_i}{\partial f_K}\boldsymbol{x_i}\\ \end{bmatrix} &= \begin{bmatrix} x_{11} & \cdots & x_{N1}\\ \vdots & \ddots & \vdots\\ x_{1D} & \cdots & x_{ND}\\ \end{bmatrix} \times \begin{bmatrix} \frac{\partial\mathcal{L}_1}{\partial f_1} & \cdots & \frac{\partial\mathcal{L}_1}{\partial f_K}\\ \vdots & \ddots & \vdots\\ \frac{\partial\mathcal{L}_N}{\partial f_1} & \cdots & \frac{\partial\mathcal{L}_N}{\partial f_K}\\ \end{bmatrix}\\ dW &= X^\top dF \end{align*}

The derivative of the loss with respect to the biases is obtained similarly. Again using the chain rule,

\begin{align} \frac{\partial\mathcal{L}_i}{\partial b_j} &= \sum_{k=1}^K \frac{\partial\mathcal{L}_i}{\partial f_k}\times\frac{\partial f_k}{\partial b_j}\\ &= \frac{\partial\mathcal{L}_i}{\partial f_j}\times\frac{\partial f_j}{\partial b_j}\\ &= \frac{\partial\mathcal{L}_i}{\partial f_j}\times 1,\\ \end{align}

which follows from the derivative $\frac{\partial f_k}{\partial \boldsymbol{b_j}} = \mathbb{1}\left(j=k\right)$. The derivative of the loss across all observations is again obtained via summation

\begin{align*} \frac{\partial\mathcal{L}}{\partial b_j} &= \sum_{i=1}^N \frac{\partial\mathcal{L}_i}{\partial b_j}\\ &= \sum_{i=1}^N \frac{\partial\mathcal{L}_i}{\partial f_j}, \end{align*}

and therefore

\begin{align} db &= \begin{bmatrix} \sum_{i=1}^N \frac{\partial\mathcal{L}_i}{\partial f_1} \\ \vdots\\ \sum_{i=1}^N \frac{\partial\mathcal{L}_i}{\partial f_K}\\ \end{bmatrix} = \begin{bmatrix} \sum_{i=1}^N dF_{i1} \\ \vdots\\ \sum_{i=1}^N dF_{iK}\\ \end{bmatrix}.\\ \end{align}