Learning from approximate data

Abstract

We give an algorithm to PAC learn the coefficients of a multivariate polynomial from the signs of its values, over a sample of real points which are only known approximately. While there are several papers dealing with PAC learning polynomials (e.g. [3, II]), they mainly only consider variables over finite fields or real variables with no round-off error. In particular, to the best of our knowledge, the only other work considering rounded-off real data is that of Dennis Cheung [6]. There, multivariate polynomials are learned under the assumption that the coefficients are independent, eventually leading to a linear programming problem. In this thesis we consider the other extreme: namely, we consider the case where the coefficients of the polynomial are (polynomial) functions of a single parameter. As we shall see, this leads to solving a non-linear system of polynomial inequalities in one variable. Our algorithm does so in a number of operations which is polynomial in the data size and the logarithm of the condition of the sample.