Often, unlabeled data is easy to come by but labels are expensive. For instance, if you’re building a speech recognizer, it’s easy enough to get raw speech samples — just walk around with a microphone — but labeling even one of these samples is a tedious process in which a human must examine the speech signal and carefully segment it into phonemes. In the field of active learning, the goal is as usual to construct an accurate classifier, but the labels of the data points are initially hidden and there is a charge for each label you want revealed. The hope is that by intelligent adaptive querying, you can get away with significantly fewer labels than you would need in a regular supervised learning framework.

Here’s an example. Suppose the data lie on the real line, and the classifiers are simple thresholding functions, H = {h_{w}}:

h_{w}(x) = 1 if x > w, and 0 otherwise.
VC theory tells us that if the underlying distribution P can be classified perfectly by some hypothesis in H (called the *realizable* case), then in order to get a classifier with error rate at most *e*, it is enough to draw *m = O(1/e)* random labeled examples from P, and to return any classifier consistent with them. But suppose we instead draw *m* *unlabeled* samples from P. If we lay these points down on the line, their hidden labels are a sequence of 0’s followed by a sequence of 1’s, and the goal is to discover the point *w* at which the transition occurs. This can be accomplished with a simple binary search which asks for just *log m* labels. Thus active learning gives us an exponential improvement in the number of labels needed: by adaptively querying *log m* labels, we can automatically infer the rest of them.

Unfortunately, not all that much is known beyond this toy example. To date, the single main theoretical result in the field is [FSST97]‘s analysis of the query-by-committee (QBC) learning algorithm. In their model, the learner observes a stream of unlabeled data and makes spot decisions about whether or not to ask for a point’s label. They show that if the data is drawn uniformly from the surface of the d-dimensional unit sphere, and the hidden labels correspond perfectly to a homogeneous (i.e., through the origin) linear separator from this same distribution, then it is possible to achieve generalization error *e* after seeing *O(d/e)* points and requesting just *O(d log 1/e)* labels: an exponential improvement over the usual *O(d/e)* sample complexity of learning linear separators in a supervised setting. This remarkable result is tempered somewhat by the complexity of the QBC algorithm, which involves computing volumes of intermediate version spaces.

Some recent progress on active learning: [DKM05] show how a simple variant of the perceptron update can be used to achieve these same sample complexity bounds, in the same model. [D04] shows a variety of upper and lower bounds for active learning — for instance, if you allow linear separators which are non-homogeneous then in the above model the sample complexity necessarily shoots up to *1/e*.

The theoretical terrain of active learning remains something of an unexplored wilderness. There has, however, been a lot of beautiful theory work (see [A02] for a roundup) on a related model in which the learner is allowed to synthesize query points, rather than simply choosing them from the pool of unlabeled data. This ran into some practical problems: [BL92] found that the resulting synthetic instances were often very difficult for a human to classify!

[A02] D. Angluin. Queries revisited.

[BL92] E. Baum and K. Lang. Query learning can work poorly when a human oracle is used.

[D04] S. Dasgupta. Analysis of a greedy active learning strategy.

[DKM05] S. Dasgupta, A. Kalai, and C. Monteleoni. Analysis of perceptron-based active learning.

[FSST97] Y. Freund, S. Seung, E. Shamir, and N. Tishby. Selective sampling using the query-by-committee algorithm.