Most of the time. Certain crimes (eg public intoxication) often have an arrest followed by letting the person go when they’ve sobered up, so no official prosecution but not no arrest.
Can you think of any other?
The DA's resources are limited (unless all the Republican critics want to pay more taxes!). Arrest is enough of a deterrent for minor crimes.
It has nothing to do with deterring crimes or crime rates, it is just politics.