We apply a general algorithm for merging prediction strategies (the Aggregating Algorithm) to the problem of linear regression with the square loss; our main assumption is that the response variable is bounded. It turns out that for this particular problem the Aggregating Algorithm resembles, but is slightly different from, the wellknown ridge estimation procedure. From general results about the Aggregating Algorithm we deduce a guaranteed bound on the difference between our algorithm's performance and the best, in some sense, linear regression function's performance. We show that the AA attains the optimal constant in our bound, whereas the constant attained by the ridge regression procedure in general can be 4 times worse. 1 INTRODUCTION The usual approach to regression problems is to assume that the data are generated by some stochastic mechanism and make some, typically very restrictive, assumptions about that stochastic mechanism. In recent years, however, a different approach to this kind of problems was developed (see, e.g., DeSantis et al. [2], Littlestone and Warmuth [7]): in our context, that approach sets the goal of finding an online algorithm that performs not much worse than the best regression function found off-line; in other words, it replaces the usual statistical analyses by the competitive analysis of online algorithms. DeSantis et al. [2] performed a competitive analysis of the Bayesian merging scheme for the log-loss prediction game; later Littlestone and Warmuth [7] and Vovk [10] introduced an online algorithm (called the Weighted Majority Algorithm by the Competitive Online Linear Regression 365 former authors) for the simple binary prediction game. These two algorithms (the Bayesian merging scheme and the Weighted Majority Algorithm) are special cases of the Aggregating Algorithm (AA) proposed in [9, 11]. The AA is a member of a wide family of algorithms called "multiplicative weight" or "exponential weight" algorithms. Closer to the topic of this paper, Cesa-Bianchi et al. [1) performed a competitive analysis, under the square loss, of the standard Gradient Descent Algorithm and Kivinen and Warmuth [6] complemented it by a competitive analysis of a modification of the Gradient Descent, which they call the Exponentiated Gradient Algorithm.

Support Vector (SV) Machines for pattern recognition, regression estimation and operator inversion exploit the idea of transforming into a high dimensional feature space where they perform a linear algorithm. Instead of evaluating this map explicitly, one uses Hilbert Schmidt Kernels k(x, y) which correspond to dot products of the mapped data in high dimensional space, i.e. k(x, y) ( I (x) · I (y))

The problem of time series prediction is studied within the uniform convergence framework of Vapnik and Chervonenkis. The dependence inherent in the temporal structure is incorporated into the analysis, thereby generalizing the available theory for memoryless processes. Finite sample bounds are calculated in terms of covering numbers of the approximating class, and the tradeoff between approximation and estimation is discussed. A complexity regularization approach is outlined, based on Vapnik's method of Structural Risk Minimization, and shown to be applicable in the context of mixing stochastic processes.

We study online generalized linear regression with multidimensional outputs, i.e., neural networks with multiple output nodes but no hidden nodes. We allow at the final layer transfer functions such as the softmax function that need to consider the linear activations to all the output neurons. We use distance functions of a certain kind in two completely independent roles in deriving and analyzing online learning algorithms for such tasks. We use one distance function to define a matching loss function for the (possibly multidimensional) transfer function, which allows us to generalize earlier results from one-dimensional to multidimensional outputs. We use another distance function as a tool for measuring progress made by the online updates. This shows how previously studied algorithms such as gradient descent and exponentiated gradient fit into a common framework. We evaluate the performance of the algorithms using relative loss bounds that compare the loss of the online algoritm to the best off-line predictor from the relevant model class, thus completely eliminating probabilistic assumptions about the data.

Recent theoretical results for pattern classification with thresholded real-valued functions (such as support vector machines, sigmoid networks, and boosting) give bounds on misclassification probability that do not depend on the size of the classifier, and hence can be considerably smaller than the bounds that follow from the VC theory. In this paper, we show that these techniques can be more widely applied, by representing other boolean functions as two-layer neural networks (thresholded convex combinations of boolean functions).

Each chart is primarily divided into the three major noise conditions, cf.

In this paper, we present a novel hybrid architecture for continuous speech recognition systems. It consists of a continuous HMM system extended by an arbitrary neural network that is used as a preprocessor that takes several frames of the feature vector as input to produce more discriminative featurevectors with respect to the underlying HMM system. This hybrid system is an extension of a state-of-the-art continuous HMM system, andin fact, it is the first hybrid system that really is capable ofoutperforming thesestandard systems with respect to the recognition accuracy. Experimental results show an relative error reduction of about 10% that we achieved on a remarkably good recognition system based on continuous HMMsfor the Resource Management 1OOO-word continuous speech recognition task.

Each chart is primarily divided into the three major noise conditions, cf.

Active data clustering is a novel technique for clustering of proximity datawhich utilizes principles from sequential experiment design in order to interleave data generation and data analysis. The proposed activedata sampling strategy is based on the expected value of information, a concept rooting in statistical decision theory. This is considered to be an important step towards the analysis of largescale datasets, because it offers a way to overcome the inherent data sparseness of proximity data.

Voice: (818) 306-6144 FAX: (818) 306-6912 Content Areas: Applications (Stochastic Optimization),Model Selection Algorithms Abstract This paper considers the problem of learning the ranking of a set of alternatives based upon incomplete information (e.g., a limited number of observations). We describe two algorithms for hypothesis rankingand their application for probably approximately correct (PAC)and expected loss (EL) learning criteria. Empirical results are provided to demonstrate the effectiveness of these ranking procedureson both synthetic datasets and real-world data from a spacecraft design optimization problem. 1 INTRODUCTION In many learning applications, the cost of information can be quite high, imposing a requirement that the learning algorithms glean as much usable information as possible with a minimum of data. For example: - In speedup learning, the expense of processing each training example can be significant [Tadepalli921. This paper provides a statistical decision-theoretic framework for the ranking of parametric distributions.