An optimization in machine learning is achieved using Newton's algorithm together with an efficient technique for solving linear equations, such as the method of conjugate residuals. The techniques of the present invention are applicable to learning language models, predicting classes of objects from images and videos, and classifying financial transactions for prevention of fraud. Other uses include determining a function from a sequence of words to a relevant web page for a search engine, or to inverting arbitrary output values of an analyzed system into an internally running simulation.