Abstract not found

We construct data dependent upper bounds on the risk in function learning problems. The bounds are based on the local norms of the Rademacher process indexed by the underlying function class and they do not require prior knowledge about the distribution of training examples or any specific properties of the function class. Using Talagrand's type concentration inequalities for empirical and Rademacher processes, we show that the bounds hold with high probability that decreases exponentially fast when the sample size grows. In typical situations that are frequently encountered in the theory of function learning, the bounds give nearly optimal rate of convergence of the risk to zero. 1. Local Rademacher norms and bounds on the risk: main results Let (S; A) be a measurable space and let F be a class of A-measurable functions from S into [0; 1]: Denote P(S) the set of all probability measures on (S; A): Let f 0 2 F be an unknown target function. Given a probability measure P 2 P(S) (also unknown), let (X 1 ; : : : ; Xn ) be an i.i.d. sample in (S; A) with common distribution P (defined on a probability space(\Omega ; \Sigma; P)). In computer learning theory, the problem of estimating f 0 ; based on the labeled sample (X 1 ; Y 1 ); : : : ; (Xn ; Yn ); where Y j := f 0 (X j ); j = 1; : : : ; n; is referred to as function learning problem. The so called concept learning is a special case of function learning. In this case, F := fI C : C 2 Cg; where C ae A is called a class of concepts (see Vapnik (1998), Vidyasagar (1996), Devroye, Gyorfi and Lugosi (1996) for the account on statistical learning theory). The goal of function learning is to find an estimate

In this article, model selection via penalized empirical loss minimization in nonparametric classification problems is studied. Datadependent penalties are constructed, which are based on estimates of the complexity of a small subclass of each model class, containing only those functions with small empirical loss. The penalties are novel since those considered in the literature are typically based on the entire model class. Oracle inequalities using these penalties are established, and the advantage of the new penalties over those based on the complexity of the whole model class is demonstrated.

Penalization procedures often suffer from their dependence on multiplying factors, whose optimal values are either unknown or hard to estimate from data. We propose a completely data-driven calibration algorithm for these parameters in the least-squares regression framework, without assuming a particular shape for the penalty. Our algorithm relies on the concept of minimal penalty, recently introduced by Birgé and Massart (2007) in the context of penalized least squares for Gaussian homoscedastic regression. On the positive side, the minimal penalty can be evaluated from the data themselves, leading to a data-driven estimation of an optimal penalty which can be used in practice; on the negative side, their approach heavily relies on the homoscedastic Gaussian nature of their stochastic framework. The purpose of this paper is twofold: stating a more general heuristics for designing a datadriven penalty (the slope heuristics) and proving that it works for penalized least-squares regression with a random design, even for heteroscedastic non-Gaussian data. For technical reasons, some exact mathematical results will be proved only for regressogram bin-width selection. This is at least a first step towards further results, since the approach and the method that we use are indeed general.

Rademacher penalization is a modern technique for obtaining data-dependent bounds on the generalization error of classi ers. It would appear 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 prunings of a given decision tree rather than the tree growing phase. Moreover, we generalize the error-bounding approach from binary classi cation to multi-class situations. Our empirical experiments indicate that the proposed new bounds clearly outperform earlier bounds for decision tree prunings and provide non-trivial error estimates on real-world data sets.

We give a permutation approach to validation (estimation of out-sample error). One typical use of validation is model selection. We establish the legitimacy of the proposed permutation complexity by proving a uniform bound on the out-sample error, similar to a VC-style bound. We extensively demonstrate this approach experimentally on synthetic data, standard data sets from the UCI-repository, and a novel diffusion data set. The out-of-sample error estimates are comparable to cross validation (CV); yet, the method is more efficient and robust, being less susceptible to overfitting during model selection. 1

The need to discretize a numerical range into class coherent intervals is a problem frequently encountered. Training Set Error (TSE) is one of the commonly used impurity functions in this task.

As compared to classical distribution-independent bounds based on the VC dimension, recent data-dependent bounds based on Rademacher complexity yield tighter upper bounds that may offer practical utility for model selection, as suggested by several investigations. We present an approach for kernel machine learning and generalization performance assessment that integrates concepts from prior work on Rademacher-type data-dependent generalization bounds and learning based on the optimization of quasiconvex losses. Our main contribution focuses on the direct estimation of the Rademacher penalty in order to obtain a tighter generalization bound. Specifically we define the optimization task for the case of learning with the ramp loss and show that direct estimation of the Rademacher penalty can be accomplished by solving a series of quadratic programming problems. 1.

We define a data dependent permutation complexity for a hypothesis set H, which is similar to a Rademacher complexity or maximum discrepancy. The crucial difference is that it is based on permutation (dependent) sampling. We prove a uniform bound on the generalization error, as well as a concentration result which means that the permutation estimate can be efficiently estimated. 1