Generating the right target for an LSTM model
The way you framed it is not a regression problem since regression predictions are unbounded. Your actual scores are strictly bounded between zero and one which makes it more similar to predicting a probability.
To get the best predictions, it would best to apply a sigmoid to the final output node activation to bound predictions between zero and one.
For a scoring, cross-entropy would work well as a loss function:
$$H(p,q) = - \sum_{i}p_i logq_i$$
The true label for a datapoint would be modeled as $p_i$ and the predicted value of the current model would be modeled as $q_i$.