We develop a new machine learning framework (theory and application) allowing one to deal with arbitrary error potentials of not-faster than quadratic growth, imitated by piece-wise quadratic function of subquadratic growth (PQSQ error potential). This universal framework is able to deal with a large family of error potentials. We exploit the fact that finding a minimum of a piece-wise quadratic function, or, in other words, a function which is the minorant of a set of quadratic functionals, can be almost as computationally efficient as optimizing the standard quadratic potential. The theory of PQSQ potentials uses {min,+} algebras and can be considered as a part of tropical mathematics.