# Visualisation of blame

I sometimes want to blame something or somebody else for errors.

This time I want to blame the inputs of neural network for its bad prediction.

The key idea is simple:

1. Imagine you have build and trained a neural network with a last layer representing $P(y \mid x)$ distribution for each input $x$.
2. Suppose that for $x_k$ input vector of the neural network predicts $\hat{y_k} \mid \hat{y_k} \neq y_{k-gold}$ which is not equal to known truth $y_k$
3. You are am not satisfied.
• I blame the neural network. Which part of the input vector does the neural network need to change so it would predict the correct input?
4. Let’s start to compute the “blame”:
1. Compute the expected $E_{x \in X}(P(y \mid x))$ distribution over your set $X$
• The expected distribution is our uninformative prior before presenting input $x_k$ but after training the network with inputs from $X$.
2. Compute gradients of the network with respect to its parameters from the loss function $\nabla_{\phi} distance(E_{x \in X}(P(y \mid x)), P(y_{k-gold} \mid x_k))$
• Note: I intend to use cross-entropy as the distance function
• The difference between the expected distribution and the gold (one-hot) distribution is how surprising the gold label is.
3. For each coordinate $x^i_k \mid i \in \{1, ..., \mid x_k \mid \}$ in input vector $x_k$ and corresponding true label sum the gradients computed from previous step $\sum_{j \in \mid FirstHidden \mid}{\nabla_{\phi} W^{i,j}_k}$
• Note that we are only interested in gradients on weights $W$ between input and first hidden layer
4. Find the member $i \in \{1, ..., \mid x_k \mid \}$ with the “largest gradient” to be blamed the most!
• Why? Suppose that our input $x_k$ consisted from words $w_1, w_2, w_3, ..., w_{\mid x_k \mid}$ and $w_2$ has the “largest gradient”. As a result, if I could change only one word in the input in order change the output of the network to the correct label, I would change $w_2$. Obviously, the network might still predict the same incorrect label or other incorrect label after changing one input. However, if there exist input $x_{artificial}$ for which the network with parameters $\phi$ predicts label $y_{k-gold}$, I would start changing values of $x_k$ into $x_{artificial}$ according the order “gradients” on the input from step 3.

I want to code it up soon, so please let me your opinions. Before that I have some todos:

1. The idea is already implemented! See Inverting a neural net
• Training the inversion directly may be easier than sampling from the expected probability over training data
5. How can I use samples from inputs distribution given the golden label $y_{k-gold}$, the parameters $\phi$ and the expected distribution $E_{x \in X}(P(y \mid x))$?