We give explicit, polynomial–time computable formulas for the number of integer points in any two– dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind–Rademacher sums, which are polynomial–time computable finite Fourier series. As a by–product we rederive a reciprocity law for these...

Rademacher penalization is a modern technique for obtaining data-dependent bounds on the generalization error of classifiers. It appears to be limited to relatively simple hypothesis classes because of computational complexity issues. In this paper we, nevertheless, apply Rademacher penalization to the in practice important hypothesis class of unrestricted decision trees by considering the prun...

This paper presents the first generalization bounds for time series prediction with a non-stationary mixing stochastic process. We prove Rademacher complexity learning bounds for both average-path generalization with non-stationary β-mixing processes and path-dependent generalization with non-stationary φ-mixing processes. Our guarantees are expressed in terms of βor φ-mixing coefficients and a...

This paper presents the first Rademacher complexity-based error bounds for noni.i.d. settings, a generalization of similar existing bounds derived for the i.i.d. case. Our bounds hold in the scenario of dependent samples generated by a stationary β-mixing process, which is commonly adopted in many previous studies of noni.i.d. settings. They benefit from the crucial advantages of Rademacher com...

We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein’s method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable wh...

Abstract. Transductive learning considers situations when a learner observes m labelled training points and u unlabelled test points with the final goal of giving correct answers for the test points. This paper introduces a new complexity measure for transductive learning called Permutational Rademacher Complexity (PRC) and studies its properties. A novel symmetrization inequality is proved, wh...