When you run a compiler, do you "host" it, or do you run it? Inference is the lingo in the ML world for running input through a model to get some output, where that execution occurs is irrelevant to the idea of inference itself.
If you have a hosted compiler service and you compiled a program with it, you wouldn't say you "compiler hosted" the source code, no, you still "compiled the source code". In the same way, with ML, if it's hosted or local when you run input through the model you performed "inference", that is the act that has occurred (we can debate the term "inference", but "model hosting" is a worse term).
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When you run a compiler, do you "host" it, or do you run it? Inference is the lingo in the ML world for running input through a model to get some output, where that execution occurs is irrelevant to the idea of inference itself.
If you have a hosted compiler service and you compiled a program with it, you wouldn't say you "compiler hosted" the source code, no, you still "compiled the source code". In the same way, with ML, if it's hosted or local when you run input through the model you performed "inference", that is the act that has occurred (we can debate the term "inference", but "model hosting" is a worse term).