Is it possible to use Tensorflow or some similar library to make a model that you can efficiently train and use at the same time.
An example/use case for this would be a chat bot that you give feedback to. Somewhat like how pets learn (i.e. replicating what they just did for a reward). Or being able to add new entries or new responses they can use.
I think what you are asking is whether a model can be trained continuously without having to retrain it from scratch each time new labelled data comes in.
Answer to that is - Online models
There are models that can be trained continuously on data without worrying about training them from scratch. As per Wikipedia definition
Some examples of such algorithms are