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Is there any way to train a regression model with negative values that is more stable?

A couple things you could try: You could try normalizing your target variable, so that it's number of standard deviations from the mean, or mapped to [-1,1]. If you are using drop-out during training. ...
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YearPrediction dataset for a regression task: is it possible to evaluate a fair comparison between standard loss and a quadratic one?

The way to evaluate any supervised learning result is to pick a metric - a scoring system for the results. Ideally this metric captures key details of what properties you care about for the trained ...
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