Is it possible to train neural networks on a scored training set? Is it possible to score each sample of a training set such that a sample with score N effects the network as if N of such samples were trained on? I also am curious about the ability to use negative scores.
 A: Yes, you can incorporate this into your loss function. For example, say you're using the squared error as the loss function for a network that performs regression. The loss function would typically be:
$$\sum_{i=1}^{n} (y_i - f(x_i))^2$$
where $x_i$ and $y_i$ are the $i$th example inputs/outputs on which the network is trained, and $f(x_i)$ is the corresponding predicted output. What you want to do is weight the training examples differently. Here, you'd use a weight vector $w$, where $w_i$ is the weight assigned to the $i$th example. The loss function would be:
$$\sum_{i=1}^{n} w_i (y_i - f(x_i))^2$$
For example, if you want the network to behave as if it has seen example $j$ twice and all other examples once, then set $w_i$ to $1$ for all $i \ne j$, and set $w_j$ to $2$.
However, negative weights don't really make sense here. Update rules change network parameters to reduce the loss function. Applying a negative weight to an example would flip the sign of that example's contribution to the loss function and cause the learning algorithm to attempt to increase the error on that example.
