High multi class classification with small data set
Yeah. There have been scenarios like that. Imagenet has more than 1000 classes. Same could be said for certain cifar datasets. 

If deep learning, go with one-hot encoding for the labels.

Stick with something like multi-class AUC as metric instead of believing on Validation accuracy. 

Go with a keras DNN, then you could try advanced architectures, with the last Dense net to have 1880 neurons each representing one label. Also stick with categorical_crossentropy for loss.

Hope this helps.