Active Learning with Statistical Models
Authors: Zoubin Ghahramani, Michael I. Jordan David A. Cohn
Published in: Advances in Neural Information Processing Systems 7  (NIPS 1994) (Main Conference Track), 1994

Abstract: For many types of learners one can compute the statistically "op(cid:173) timal" way to select data. We review how these techniques have  been used with feedforward neural networks [MacKay, 1992; Cohn,  1994] . 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 regres(cid:173) sion are both efficient and accurate. 
1 ACTIVE LEARNING - BACKGROUND 
An active learning problem is one where the learner has the ability or need to  influence or select its own training data. Many problems of great practical interest  allow active learning, and many even require it.  We consider the problem of actively learning a mapping X - Y based on a set of  training examples {(Xi,Yi)}~l' where Xi E X and Yi E Y. The learner is allowed  to iteratively select new inputs x (possibly from a constrained set), observe the  resulting output y, and incorporate the new examples (x, y) into its training set.  The primary question of active learning is how to choose which x to try next.  There are many heuristics for choosing x based on intuition, including choosing  places where we don't have data, where we perform poorly [Linden and Weber,  1993], where we have low confidence [Thrun and Moller, 1992], where we expect it 
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David Cohn, Zoubin Ghahramani, Michael I. Jordon 
to change our model [Cohn et aI, 1990], and where we previously found data that  resulted in learning [Schmidhuber and Storck, 1993].  In this paper we consider how one may select x "optimally" from a statistical  viewpoint. We first review how the statistical approach can be applied to neural  networks, as described in MacKay [1992] and Cohn [1994]. We then consider two  alternative, statistically-based learning architectures: mixtures of Gaussians and  locally weighted regression. While optimal data selection for a neural network is  computationally expensive and approximate, we find that optimal data selection for  the two statistical models is efficient and accurate. 
2 ACTIVE LEARNING - A STATISTICAL APPROACH 
We denote the learner's output given input x as y(x). The mean squared error of  this output can be expressed as the sum of the learner's bias and variance. The  variance 0'3 (x) indicates the learner's uncertainty in its estimate at x. 1 Our goal  will be to select a new example x such that when the resulting example (x, y) is  added to the training set, the integrated variance IV is minimized: 
IV = J 0'3 P (x)dx. 
(1) 
Here, P(x) is the (known) distribution over X. In practice, we will compute a  Monte Carlo approximation of this integral, evaluating 0'3 at a number of random  points drawn according to P(x).  Selecting x so as to minimize IV requires computing 0-3, the new variance at x given  (x, y). Until we actually commit to an x, we do not know what corresponding y we  will see, so the minimization cannot be performed deterministically.2 Many learning  architectures, however, provide an estimate of PWlx) based on current data, so we  can use this estimate to compute the expectation of 0-3. Selecting x to minimize  the expected integrated variance provides a solid statistical basis for choosing new  examples. 
2.1 EXAMPLE: ACTIVE LEARNING WITH A NEURAL