We study the problem of learning an unknown function represented as an expression or a program over a known finite monoid. As in other areas of computational complexity where programs over algebras have been used, the goal is to relate the computational complexity of the learning problem with the algebraic complexity of the finite monoid. Indeed, our results indicate a close connection between both kinds of complexity. We present results for Abelian, nilpotent, solvable, and nonsolvable groups, as well as for some important subclasses of aperiodic monoids.