A minimizer is a point in the input space where a function reaches its lowest value in that vicinity (or potentially anywhere). Its corresponding minimum is the value of the function at that point.
Put another way, the minimum is , whereas the minimizer is . Hence, though ML practitioners often talk about, e.g., “overshooting the minimum,” they are probably visualizing the point in parameter space where that minimum is found: the corresponding minimizer.
Formally, a set of parameters for a function is a minimizer if (but only if):
The above condition is not strictly necessary: if is only positive semidefinite, may still correspond to a local minimizer. It may also correspond to a flat “valley” that is lower than all other nearby points. Rarely, it may correspond to a more exotic change in topology.