Multi-class classification with mostly zero valued data
Welcome! Sounds like your data set is highly imbalanced. For techniques on handling an imbalanced data set, please have a look at this. https://machinelearningmastery.com/tactics-to-combat-imbalanced-classes-in-your-machine-learning-dataset/

EDIT: Looks like I misunderstood the question, sorry! Also: I think the term "multi-class regression" does not exist. It sounds more like you're doing multi-class classification. If your input features are consistently 0, then it sounds like your actual value of each feature and the mean of the feature are identical, in which case you have very poor features. The features will not do a good job of differentiating between the classes. You should try (or hopefully use a neural network to learn) a new set of features.