There are no hard and fast rules. There is a major doubt about kind of problems that are to be dealt with deep learning algorithms. How deep is this deep is matter of empiricism.
Usually, it's "the more the better" - when having less instances, you need smaller models in order not to overfit, and the overall accuracy will be smaller. But networks usually learn, even with a few instances only. In any way, expect an improvement of the classification accuracy when you increase your dataset (and model) size.
It depends on your input data what's enough - best is just to try it. For comparison - a baseline MNIST handwritten digit dataset consists of 50 000 training instances for 10 classes, and this is definitely enough. Everything above 10 000 is at least not "very small" - if the data within classes is very similar, you need less - if it is very differing, you need more. Just try it. 200 000 seems enough for most problems.
I strongly recommend to you this lecture from the course "Learning From Data" provided by Caltech:
https://www.youtube.com/watch?v=Dc0sr0kdBVI
In this lecture, the concepts of the Vapnik–Chervonenkis (VC) inequality and VC dimension are explained in detail. The VC inequality is the mathematical concept for explaining how good our generalization error is regarding the number of training samples (among other variables).
Generally it is said that in deep learning, model needs more data than the conventional pattern classifiers like SVM, ANN etc. It is because in deep learning we don't use any handcrafted features, so the model learns from the patterns of raw images. Therefore, in these auto-learning models, if we have large number of image classes to to be classified then we need to provide large number of pattern samples. Otherwise, it would not perform well, when the test image will have extreme variations. We have worked in deep learning environment to classify handwritten digit recognition, and for that we have used different sets of training and testing samples (like 100, 200, 500, 1000 samples per digit class). Hope this helps.