An adaptive online algorithm is proposed to estimate hierarchical data structures for non-stationary data sources. The approach is based on the principle of minimum cross entropy to derive a decision tree for data clustering and it employs a metalearning idea (learning to learn) to adapt to changes in data characteristics. Its efficiency is demonstrated by grouping non-stationary artifical data and by hierarchical segmentation of LANDSAT images. 1 Introduction Unsupervised learning addresses the problem to detect structure inherent in unlabeled and unclassified data. N. The encoding usually is represented by an assignment matrix M (Mia), where Mia 1 if and only if Xi belongs to cluster L: 1 MiaV (Xi, Ya) measures the quality of a data partition, Le., optimal assignments and prototypes (M,y)OPt argminM,y1i (M,Y) minimize the inhomogeneity of clusters w.r.t. a given distance measure V. For reasons of simplicity we restrict the presentation to the ' sum-of-squared-error criterion V(x, y) To facilitate this minimization a deterministic annealing approach was proposed in [5] signments, which maps the discrete optimization problem, i.e. how to determine the data as via the Maximum Entropy Principle [2] to a continuous parameter es- Unsupervised Online Learning of Decision Trees for Data Analysis 515 timation problem.

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.

A simple linear averaging of the outputs of several networks as e.g. in bagging [3], seems to follow naturally from a bias/variance decomposition of the sum-squared error. The sum-squared error of the average model is a quadratic function of the weighting factors assigned to the networks in the ensemble [7], suggesting a quadratic programming algorithm for finding the "optimal" weighting factors. If we interpret the output of a network as a probability statement, the sum-squared error corresponds to minus the loglikelihood or the Kullback-Leibler divergence, and linear averaging of the outputs tologarithmic averaging of the probability statements: the logarithmic opinion pool. The crux of this paper is that this whole story about model averaging, bias/variancedecompositions, and quadratic programming to find the optimal weighting factors, is not specific for the sumsquared error,but applies to the combination of probability statements of any kind in a logarithmic opinion pool, as long as the Kullback-Leibler divergence plays the role of the error measure. As examples we treat model averaging for classification models under a cross-entropy error measure and models for estimating variances.

We are frequently called upon to perform multiple tasks that compete forour attention and resource. Often we know the optimal solution to each task in isolation; in this paper, we describe how this knowledge can be exploited to efficiently find good solutions for doing the tasks in parallel. We formulate this problem as that of dynamically merging multiple Markov decision processes (MDPs) into a composite MDP, and present a new theoretically-sound dynamic programmingalgorithm for finding an optimal policy for the composite MDP. We analyze various aspects of our algorithm and illustrate its use on a simple merging problem. Every day, we are faced with the problem of doing mUltiple tasks in parallel, each of which competes for our attention and resource. If we are running a job shop, we must decide which machines to allocate to which jobs, and in what order, so that no jobs miss their deadlines. If we are a mail delivery robot, we must find the intended recipients of the mail while simultaneously avoiding fixed obstacles (such as walls) and mobile obstacles (such as people), and still manage to keep ourselves sufficiently charged up. Frequently we know how to perform each task in isolation; this paper considers how we can take the information we have about the individual tasks and combine it to efficiently find an optimal solution for doing the entire set of tasks in parallel. More importantly, we describe a theoretically-sound algorithm for doing this merging dynamically; new tasks (such as a new job arrival at a job shop) can be assimilated online into the solution being found for the ongoing set of simultaneous tasks.

An adaptive online algorithm is proposed to estimate hierarchical data structures for non-stationary data sources. The approach is based on the principle of minimum cross entropy to derive a decision tree for data clustering and it employs a metalearning idea (learning to learn) to adapt to changes in data characteristics. Its efficiency is demonstrated by grouping non-stationary artifical data and by hierarchical segmentation of LANDSAT images. 1 Introduction Unsupervised learning addresses the problem to detect structure inherent in unlabeled andunclassified data. N. The encoding usually is represented by an assignment matrix M (Mia), where Mia 1 if and only if Xi belongs to cluster L: 1 MiaV (Xi, Ya) measures the quality of a data partition, Le., optimal assignments and prototypes (M,y)OPt argminM,y1i (M,Y) minimize the inhomogeneity of clusters w.r.t. a given distance measure V. For reasons of simplicity we restrict the presentation to the ' sum-of-squared-error criterion V(x, y) To facilitate this minimization a deterministic annealing approach was proposed in [5] which maps the discrete optimization problem, i.e. how to determine the data assignments, viathe Maximum Entropy Principle [2] to a continuous parameter es- Unsupervised Online Learning ofDecision Trees for Data Analysis 515 timation problem.

This paper describes some of the interactions of model learning algorithms and planning algorithms we have found in exploring model-based reinforcement learning. The paper focuses on how local trajectoryoptimizers can be used effectively with learned nonparametric models.We find that trajectory planners that are fully consistent with the learned model often have difficulty finding reasonable plansin the early stages of learning. Trajectory planners that balance obeying the learned model with minimizing cost (or maximizing reward) often do better, even if the plan is not fully consistent with the learned model. 1 INTRODUCTION We are exploring the use of nonparametric models in robot learning (Atkeson et al., 1997b; Atkeson and Schaal, 1997). This paper describes the interaction of model learning algorithms and planning algorithms, focusing on how local trajectory optimization canbe used effectively with nonparametric models in reinforcement learning. We find that trajectory optimizers that are fully consistent with the learned model often have difficulty finding reasonable plans in the early stages of learning. The message of this paper is that a planner should not be entirely consistent with the learned model during model-based reinforcement learning.

In many optimization problems, the structure of solutions reflects complex relationships between the different input parameters. For example, experience may tell us that certain parameters are closely related and should not be explored independently. Similarly, experience may establish that a subset of parameters must take on particular values. Any search of the cost landscape should take advantage of these relationships. We present MIMIC, a framework in which we analyze the global structure of the optimization landscape. A novel and efficient algorithm for the estimation of this structure is derived. We use knowledge of this structure to guide a randomized search through the solution space and, in turn, to refine our estimate ofthe structure.

The genetic algorithm (GA) is a heuristic search procedure based on mechanisms abstracted from population genetics. In a previous paper [Baluja & Caruana, 1995], we showed that much simpler algorithms, such as hillcIimbing and Population Based Incremental Learning (PBIL), perform comparably to GAs on an optimization problem custom designed to benefit from the GA's operators. This paper extends these results in two directions. First, in a large-scale empirical comparison of problems that have been reported in GA literature, we show that on many problems, simpler algorithms can perform significantly better than GAs. Second, we describe when crossover is useful, and show how it can be incorporated into PBIL. 1 IMPLICIT VS.

There are many approaches to this problem, including statistical [9], machine learning [7], integer and mathematical programming [18,1]. In this paper we concentrate on a simple concave minimization formulation of the problem that leads to a finite and fast algorithm.

When triangulating a belief network we aim to obtain a junction tree of minimum state space. According to (Rose, 1970), searching for the optimal triangulation can be cast as a search over all the permutations of the graph's vertices. Our approach is to embed the discrete set of permutations in a convex continuous domain D. By suitably extending the cost function over D and solving the continous nonlinear optimization task we hope to obtain a good triangulation with respect to the aformentioned cost. This paper presents two ways of embedding the triangulation problem into continuous domain and shows that they perform well compared to the best known heuristic.