Theorem 1 only provides conditions under which the deviation between erP (h) and ^erz (h) is likely to be small, it does not guarantee that the true error erP (h) will actually be small. This is governed by the choice of H. If H contains a solution with small error and the learner minimizes error on the training set, then with high probability erP (h) will be small. However, a bad choice of H will mean there is no hope of achieving small error. Thus, the *bias* of the learner in this model2 is represented by the choice of hypothesis space H.

## 2.2 The Bias Learning Model

The main extra assumption of the bias learning model introduced here is that the learner is embedded in an *environment* of related tasks, and can sample from the environment to generate multiple training sets belonging to multiple different tasks. In the above model of ordinary (single-task) learning, a learning task is represented by a distribution P on X - Y . So in the bias learning model, an environment of learning problems is represented by a pair (P; Q) where P is the set of all probability distributions on X Y (i.e., P is the set of all possible learning problems), and Q is a distribution on P. Q controls which learning problems the learner is likely to see 3. For example, if the learner is in a face recognition environment, Q will be highly peaked over face-recognition-type problems, whereas if the learner is in a character recognition environment Q will be peaked over character-recognition-type problems (here, as in the introduction, we view these environments as sets of individual classification problems, rather than single, multiple class classification problems).

Recall from the last paragraph of the previous section that the learner's bias is represented by its choice of hypothesis space H. So to enable the learner to learn the bias, we supply it with a *family* or set of hypothesis spaces H := fHg.

Putting all this together, formally a learning to learn or *bias learning* problem consists of:
 an *input space* X and an *output space* Y (both of which are separable metric spaces),
 a *loss function* l : Y - Y ! R,

 an *environment* (P; Q) where P is the set of all probability distributions on X - Y and Q is a distribution on P,
 a *hypothesis space family* H = fHg where each H 2 H is a set of functions h : X ! Y .

From now on we will assume the loss function l has range [0; 1℄, or equivalently, with rescaling, we assume that l is bounded.