Reinforcement Learning by Policy Search

One objective of artificial intelligence is to model the behavior of an intelligent agent interacting with its environment. The environment's transformations can 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. This 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 is constrained by the architecture of the agent's controller. POMDPs require a controller to have a memory. We investigate controllers with memory, including controllers with external memory, finite state controllers and distributed controllers for multi-agent systems. 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.

A Note on the Generalization Performance of Kernel Classifiers with Margin

We present distribution independent bounds on the generalization misclassification performance of a family of kernel classifiers with margin. Support Vector Machine classifiers (SVM) stem out of this class of machines. The bounds are derived through computations of the $V_gamma$ dimension of a family of loss functions where the SVM one belongs to. Bounds that use functions of margin distributions (i.e. functions of the slack variables of SVM) are derived.

Comparing Support Vector Machines with Gaussian Kernels to Radial Basis Function Classifiers

The Support Vector (SV) machine is a novel type of learning machine, based on statistical learning theory, which contains polynomial classifiers, neural networks, and radial basis function (RBF) networks as special cases. In the RBF case, the SV algorithm automatically determines centers, weights and threshold such as to minimize an upper bound on the expected test error. The present study is devoted to an experimental comparison of these machines with a classical approach, where the centers are determined by $k$--means clustering and the weights are found using error backpropagation. We consider three machines, namely a classical RBF machine, an SV machine with Gaussian kernel, and a hybrid system with the centers determined by the SV method and the weights trained by error backpropagation. Our results show that on the US postal service database of handwritten digits, the SV machine achieves the highest test accuracy, followed by the hybrid approach. The SV approach is thus not only theoretically well--founded, but also superior in a practical application.

Neural Networks

We present an overview of current research on artificial neural networks, emphasizing a statistical perspective. We view neural networks as parameterized graphs that make probabilistic assumptions about data, and view learning algorithms as methods for finding parameter values that look probable in the light of the data. We discuss basic issues in representation and learning, and treat some of the practical issues that arise in fitting networks to data. We also discuss links between neural networks and the general formalism of graphical models.

A Note on Support Vector Machines Degeneracy

When training Support Vector Machines (SVMs) over non-separable data sets, one sets the threshold $b$ using any dual cost coefficient that is strictly between the bounds of $0$ and $C$. We show that there exist SVM training problems with dual optimal solutions with all coefficients at bounds, but that all such problems are degenerate in the sense that the "optimal separating hyperplane" is given by ${f w} = {f 0}$, and the resulting (degenerate) SVM will classify all future points identically (to the class that supplies more training data). We also derive necessary and sufficient conditions on the input data for this to occur. Finally, we show that an SVM training problem can always be made degenerate by the addition of a single data point belonging to a certain unboundedspolyhedron, which we characterize in terms of its extreme points and rays.

The Informational Complexity of Learning from Examples

This thesis attempts to quantify the amount of information needed to learn certain tasks. The tasks chosen vary from learning functions in a Sobolev space using radial basis function networks to learning grammars in the principles and parameters framework of modern linguistic theory. These problems are analyzed from the perspective of computational learning theory and certain unifying perspectives emerge.

Sequential Optimal Recovery: A Paradigm for Active Learning

In most classical frameworks for learning from examples, it is assumed that examples are randomly drawn and presented to the learner. In this paper, we consider the possibility of a more active learner who is allowed to choose his/her own examples. Our investigations are carried out in a function approximation setting. In particular, using arguments from optimal recovery (Micchelli and Rivlin, 1976), we develop an adaptive sampling strategy (equivalent to adaptive approximation) for arbitrary approximation schemes. We provide a general formulation of the problem and show how it can be regarded as sequential optimal recovery. We demonstrate the application of this general formulation to two special cases of functions on the real line 1) monotonically increasing functions and 2) functions with bounded derivative. An extensive investigation of the sample complexity of approximating these functions is conducted yielding both theoretical and empirical results on test functions. Our theoretical results (stated insPAC-style), along with the simulations demonstrate the superiority of our active scheme over both passive learning as well as classical optimal recovery. The analysis of active function approximation is conducted in a worst-case setting, in contrast with other Bayesian paradigms obtained from optimal design (Mackay, 1992).

Learning from Incomplete Data

Real-world learning tasks often involve high-dimensional data sets with complex patterns of missing features. In this paper we review the problem of learning from incomplete data from two statistical perspectives---the likelihood-based and the Bayesian. The goal is two-fold: to place current neural network approaches to missing data within a statistical framework, and to describe a set of algorithms, derived from the likelihood-based framework, that handle clustering, classification, and function approximation from incomplete data in a principled and efficient manner. These algorithms are based on mixture modeling and make two distinct appeals to the Expectation-Maximization (EM) principle (Dempster, Laird, and Rubin 1977)---both for the estimation of mixture components and for coping with the missing data.

A Unified Framework for Regularization Networks and Support Vector Machines

Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples -- in particular the regression problem of approximating a multivariate function from sparse data. We present both formulations in a unified framework, namely in the context of Vapnik's theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics.

A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.

Extensions of a Theory of Networks for Approximation and Learning: Outliers and Negative Examples

Learning an input-output mapping from a set of examples can be regarded as synthesizing an approximation of a multi-dimensional function. From this point of view, this form of learning is closely related to regularization theory. In this note, we extend the theory by introducing ways of dealing with two aspects of learning: learning in the presence of unreliable examples and learning from positive and negative examples. The first extension corresponds to dealing with outliers among the sparse data. The second one corresponds to exploiting information about points or regions in the range of the function that are forbidden.

Active Learning with Statistical Models

For many types of learners one can compute the statistically 'optimal' way to select data. We review how these techniques have been used with feedforward neural networks. We then show how the same principles may be used to select data for two alternative, statistically-based learning architectures: mixtures of Gaussians and locally weighted regression. While the techniques for neural networks are expensive and approximate, the techniques for mixtures of Gaussians and locally weighted regression are both efficient and accurate.

Mean Field Theory for Sigmoid Belief Networks

We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition -- the classification of handwritten digits.

A Unified Statistical and Information Theoretic Framework for Multi-modal Image Registration

We formulate and interpret several multi-modal registration methods in the context of a unified statistical and information theoretic framework. A unified interpretation clarifies the implicit assumptions of each method yielding a better understanding of their relative strengths and weaknesses. Additionally, we discuss a generative statistical model from which we derive a novel analysis tool, the "auto-information function", as a means of assessing and exploiting the common spatial dependencies inherent in multi-modal imagery. We analytically derive useful properties of the "auto-information" as well as verify them empirically on multi-modal imagery. Among the useful aspects of the "auto-information function" is that it can be computed from imaging modalities independently and it allows one to decompose the search space of registration problems.