Dedicated to A.V. Skorohod on his seventieth birthday We prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers. Such combinations could be implemented by neural networks or by voting methods of combining the classifiers, such as boosting and bagging. The bounds are in terms of the empirical distribution of the margin of the combined classifier. They are based on the methods of the theory of Gaussian and empirical processes (comparison inequalities, symmetrization method, concentration inequalities) and they improve previous results of Bartlett (1998) on bounding the generalization error of neural networks in terms of ℓ1-norms of the weights of neurons and of Schapire, Freund, Bartlett and Lee (1998) on bounding the generalization error of boosting. We also obtain rates of convergence in Lévy distance of empirical margin distribution to the true margin distribution uniformly over the classes of classifiers and prove the optimality of these rates.

We propose new bounds on the error of learning algorithms in terms of a data-dependent notion of complexity. The estimates we establish give optimal rates and are based on a local and empirical version of Rademacher averages, in the sense that the Rademacher averages are computed from the data, on a subset of functions with small empirical error. We present some applications to classification and prediction with convex function classes, and with kernel classes in particular.

Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 0-1 loss function. The convexity makes these algorithms computationally efficient. The use of a surrogate, however, has statistical consequences that must be balanced against the computational virtues of convexity. To study these issues, we provide a general quantitative relationship between the risk as assessed using the 0-1 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function: that it satisfy a pointwise form of Fisher consistency for classification. The relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the case of low noise. Finally, we

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

Abstract not found

One objective of artificial intelligence is to model the behavior of an intelligent agent interacting with its environment. The environment's transformations could be modeled as a Markov chain, whose state is partially observable to the agent and affected by its actions; such processes are known as partially observable Markov decision processes (POMDPs). While the environment's dynamics are assumed to obey certain rules, the agent does not know them and must learn. In this dissertation we focus on the agent's adaptation as captured by the reinforcement learning framework. Reinforcement learning means learning a policy---a mapping of observations into actions---based on feedback from the environment. The learning can be viewed as browsing a set of policies while evaluating them by trial through interaction with the environment. The set of policies being searched is constrained by the architecture of the agent's controller. POMDPs require a controller to have a memory. We investigate various architectures for controllers with memory, including controllers with external memory, finite state controllers and distributed controllers for multi-agent system. For these various controllers we work out the details of the algorithms which learn by ascending the gradient of expected cumulative reinforcement. Building on statistical learning theory and experiment design theory, a policy evaluation algorithm is developed for the case of experience re-use. We address the question of sufficient experience for uniform convergence of policy evaluation and obtain sample complexity bounds for various estimators. Finally, we demonstrate the performance of the proposed algorithms on several domains, the most complex of which is simulated adaptive packet routing in a telecommunication network.

Abstract. Concentration inequalities deal with deviations of functions of independent random variables from their expectation. In the last decade new tools have been introduced making it possible to establish simple and powerful inequalities. These inequalities are at the heart of the mathematical analysis of various problems in machine learning and made it possible to derive new efficient algorithms. This text attempts to summarize some of the basic tools. 1

We consider a generalization of Costa's writing on dirty paper model in which a powerlimited communicator encounters two independent sources of additive noise, one of which is known non-causally to the encoder. We seek to characterize when the capacity for this channel would not change if the source of noise known to the encoder were also known to the decoder; we call this property private-public equivalence (PPE). Costa showed that this model has PPE if both sources of noise are IID Gaussian. We show that this model has PPE as long as the unknown noise is Gaussian (but not necessarily IID) for any distribution on the known noise.

Abstract. We investigate measures of complexity of function classes based on continuity moduli of Gaussian and Rademacher processes. For Gaussian processes, we obtain bounds on the continuity modulus on the convex hull of a function class in terms of the same quantity for the class itself. We also obtain new bounds on generalization error in terms of localized Rademacher complexities. This allows us to prove new results about generalization performance for convex hulls in terms of characteristics of the base class. As a byproduct, we obtain a simple proof of some of the known bounds on the entropy of convex hulls.