Foundations of Data Science

6.14.2 Active learning
Active learning refers to algorithms that take an active role in the selection of which examples
are labeled. The algorithm is given an initial unlabeled set U of data points drawn
from distribution D and then interactively requests for the labels of a small number of
these examples. The aim is to reach a desired error rate ɛ using much fewer labels than
would be needed by just labeling random examples (i.e., passive learning).
As a simple example, suppose that data consists of points on the real line and H =
{f a : f a (x) = 1 iff x ≥ a} for a ∈ R. That is, H is the set of all threshold functions on
the line. It is not hard to show (see Exercise 6.2) that a random labeled sample of size
O( 1 log( 1 )) is sufficient to ensure that with probability ≥ 1 − δ, any consistent threshold
ɛ δ
a ′ has error at most ɛ. Moreover, it is not hard to show that Ω( 1 ) random examples are
ɛ
necessary for passive learning. However, with active learning we can achieve error ɛ using
only O(log( 1) + log log( 1 )) labels. Specifically, first draw an unlabeled sample U of size
ɛ δ
O( 1 log( 1 )). Then query the leftmost and rightmost points: if these are both negative
ɛ δ
then output a ′ = ∞, and if these are both positive then output a ′ = −∞. Otherwise (the
leftmost is negative and the rightmost is positive), perform binary search to find two adjacent
examples x, x ′ such that x is negative and x ′ is positive, and output a ′ = (x+x ′ )/2.
This threshold a ′ is consistent with the labels on the entire set U, and so by the above
argument, has error ≤ ɛ with probability ≥ 1 − δ.
The agnostic case, where the target need not belong in the given class H is quite a bit
more subtle, and addressed in a quite general way in the “A 2 ” Agnostic Active learning
algorithm [?]. For more information on active learning, see [?, ?].
6.14.3 Multi-task learning
In this chapter we have focused on scenarios where our goal is to learn a single target
function c ∗ . However, there are also scenarios where one would like to learn multiple target
functions c ∗ 1, c ∗ 2, . . . , c ∗ n. If these functions are related in some way, then one could hope to
do so with less data per function than one would need to learn each function separately.
This is the idea of multi-task learning.
One natural example is object recognition. Given an image x, c ∗ 1(x) might be 1 if x
is a coffee cup and 0 otherwise; c ∗ 2(x) might be 1 if x is a pencil and 0 otherwise; c ∗ 3(x)
might be 1 if x is a laptop and 0 otherwise. These recognition tasks are related in that
image features that are good for one task are likely to be helpful for the others as well.
Thus, one approach to multi-task learning is to try to learn a common representation
under which each of the target functions can be described as a simple function. Another
natural example is personalization. Consider a speech recognition system with n different
users. In this case there are n target tasks (recognizing the speech of each user) that are
clearly related to each other. Some good references for multi-task learning are [?, ?].
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