Abstract. Active learning differs from "learning from examples " in that the learning algorithm assumes at least some control over what part of the input domain it receives information about. In some situations, active learning is provably more powerful than learning from examples alone, giving better generalization for a fixed number of training examples. In this article, we consider the problem of learning a binary concept in the absence of noise. We describe a formalism for active concept learning called selective sampling and show how it may be approximately implemented by a neural network. In selective sampling, a learner receives distribution information from the environment and queries an oracle on parts of the domain it considers "useful. " We test our implementation, called an SGnetwork, on three domains and observe significant improvement in generalization.

We propose a new approach to the problem of searching a space of policies for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP), given a model. Our approach is based on the following observation: Any (PO)MDP can be transformed into an "equivalent" POMDP in which all state transitions (given the current state and action) are deterministic. This reduces the general problem of policy search to one in which we need only consider POMDPs with deterministic transitions. We give a natural way of estimating the value of all policies in these transformed POMDPs. Policy search is then simply performed by searching for a policy with high estimated value. We also establish conditions under which our value estimates will be good, recovering theoretical results similar to those of Kearns, Mansour and Ng [7], but with "sample complexity" bounds that have only a polynomial rather than exponential dependence on the horizon time. Our method appl...

We consider the problem of choosing a near-best strategy from a restricted class of strategies in a partially observable Markov decision process (POMDP). We assume we are given the ability to simulate the behavior of the POMDP, and we provide methods for generating simulated experience su cient to accurately approximate the expected return of any strategy in the class. We prove upper bounds on the amount of simulated experience our methods must generate in order to achieve such uniform approximation. These bounds have no dependence on the size or complexity of the underlying POMDP, but depend only on the complexity of the restricted strategy class. The main challenge is in generating trajectories in the POMDP that can be reused, in the sense that they simultaneously provide estimates of the return of many strategies in the class. Our measure of strategy class complexity generalizes the classical notion of VC dimension, and our methods develop connections between problems of current interest in reinforcement learning and well-studied issues in the theory of supervised learning. We also discuss a number of practical planning algorithms for POMDPs that arise from our reusable trajectories.

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Abstract In this paper we introduce and investigate a mathematically rigorous theory of learning curves that is based on ideas from statistical mechanics. The advantage of our theory over the well-established Vapnik-Chervonenkis theory is that our bounds can be considerably tighter in many cases, and are also more reflective of the true behavior (functional form) of learning curves. This behavior can often exhibit dramatic properties such as phase transitions, as well as power law asymptotics not explained by the VC theory. The disadvantages of our theory are that its application requires knowledge of the input distribution, and it is limited so far to finite cardinality function classes. We illustrate our results with many concrete examples of learning curve bounds derived from our theory. 1 Introduction According to the Vapnik-Chervonenkis (VC) theory of learning curves [27, 26], minimizing empirical error within a function class F on a random sample of m examples leads to generalization error bounded by ~O(d=m) (in the case that the target function is contained in F) or ~O(pd=m) plus the optimal generalization error achievable within F (in the general case). 1 These bounds are universal: they hold for any class of hypothesis functions F, for any input distribution, and for any target function. The only problem-specific quantity remaining in these bounds is the VC dimension d, a measure of the complexity of the function class F. It has been shown that these bounds are essentially the best distribution-independent bounds possible, in the sense that for any function class, there exists an input distribution for which matching lower bounds on the generalization error can be given [5, 7, 22].

Abstract. We consider the problem of PAC learning probabilistic networks in the case where the structure of the net is specified beforehand. We allow the conditional probabilities to be represented in any manner (as tables or specialized functions) and obtain sample complexity bounds for learning nets with and without hidden nodes.

Abstract Sublinear time algorithms represent a new paradigm in computing, where an algorithmmust give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1 Introduction The goal of algorithmic research is to design efficient algorithms, where efficiency is typicallymeasured as a function of the length of the input. For instance, the elementary school algorithm for multiplying two n digit integers takes roughly n2 steps, while more sophisticated algorithmshave been devised which run in less than n log2 n steps. It is still not known whether a linear time algorithm is achievable for integer multiplication. Obviously any algorithm for this task, as for anyother nontrivial task, would need to take at least linear time in n, since this is what it would take to read the entire input and write the output. Thus, showing the existence of a linear time algorithmfor a problem was traditionally considered to be the gold standard of achievement. Nevertheless, due to the recent tremendous increase in computational power that is inundatingus with a multitude of data, we are now encountering a paradigm shift from traditional computational models. The scale of these data sets, coupled with the typical situation in which there is verylittle time to perform our computations, raises the issue of whether there is time to consider any more than a miniscule fraction of the data in our computations? Analogous to the reasoning thatwe used for multiplication, for most natural problems, an algorithm which runs in sublinear time must necessarily use randomization and must give an answer which is in some sense imprecise.Nevertheless, there are many situations in which a fast approximate solution is more useful than a slower exact solution.

We show that, given data from a mixture of k well-separated spherical Gaussians in R d, a simple two-round variant of EM will, with high probability, learn the parameters of the Gaussians to nearoptimal precision, if the dimension is high (d ≫ lnk). We relate this to previous theoretical and empirical work on the EM algorithm.

We compare Bayes Model Averaging, BMA, to a non-Bayes form of model averaging called stacking. In stacking, the weights are no longer posterior probabilities of models; they are obtained by a technique based on cross-validation. When the correct data generating model (DGM) is on the list of models under consideration BMA is never worse than stacking and often is demonstrably better, provided that the noise level is of order commensurate with the coefficients and explanatory variables. Here, however, we focus on the case that the correct DGM is not on the model list and may not be well approximated by the elements on the model list. We give a sequence of computed examples by choosing model lists and DGM’s to contrast the risk performance of stacking and BMA. In the first examples, the model lists are chosen to reflect geometric principles that should give good performance. In these cases, stacking typically outperforms BMA, sometimes by a wide margin. In the second set of examples we examine how stacking and BMA perform when the model list includes all subsets of a set of potential predictors. When we standardize the size of terms and coefficients in this setting, we find that BMA outperforms stacking when the deviant terms in the DGM ‘point ’ in directions accommodated by the model list but that when the deviant term points outside the model list stacking seems to do better. Overall, our results suggest the stacking has better robustness properties than BMA in the most important settings.

ii To my wife, Ma'ayan, and my daughter, Shira. iii Acknowledgments Special thanks are due to: ffl Prof. Naftali Tishby for his help and guidance in carrying out this study, for the many fascinating discussions we had, and for the immense body of knowledge that I have absorbed from him during my studies.