Updated answer
As several comments below mention, the situation is not as simple as I originally wrote in 2013.
The generally accepted view is that
- weak supervision - supervision with noisy labels (wikipedia)
- semi supervision - only a subset of training data has labels (wikipedia)
There are also classifications that are more along with my original answer, for example, Zhi-Hua Zhou's 2017 A brief introduction to weakly supervised learning considers weak supervision to be an umbrella term for
- incomplete supervision - only a subset of training data has labels (same as above)
- inexact supervision - called where the training data are given with only coarse-grained labels
- inaccurate supervision - where the given labels are not always ground-truth (weak supervision above).
Original answer
In short: In weakly supervised learning, you use a limited amount of labeled data.
How you select this data, and what exactly you do with it depends on the method. In general you use a limited number of data that is easy to get and/or makes a real difference and then learn the rest. I consider bootstrapping to be a method that can be used in weakly supervised learning, but as the comment by Ben below shows, this is not a generally accepted view.
See, for example Chris Bieman's 2007 dissertation for a nice overview, it says the following about bootstrapping/weakly-supervised learning:
Bootstrapping, also called self-training, is a form of learning that
is designed to use even less training examples, therefore sometimes
called weakly-supervised. Bootstrapping starts with a few training
examples, trains a classifier, and uses thought-to-be positive
examples as yielded by this classifier for retraining. As the set of
training examples grows, the classifier improves, provided that not
too many negative examples are misclassified as positive, which could
lead to deterioration of performance.
For example, in case of part-of-speech tagging, one usually trains an HMM (or maximum-entropy or whatever) tagger on 10,000's words, each with it's POS. In the case of weakly supervised tagging, you might simply use a very small corpus of 100s words. You get some tagger, you use it to tag a corpus of 1000's words, you train a tagger on that and use it to tag even bigger corpus. Obviously, you have to be smarter than this, but this is a good start. (See this paper for a more advance example of a bootstrapped tagger)
Note: weakly supervised learning can also refer to learning with noisy labels (such labels can but do not need to be the result of bootstrapping)