# Weak Teacher

Weak teachers represent a class of prediction algorithms for the case of the relaxation of the on-line protocol - the case when Reality provides true labels of examples with a delay or only occasionally, for a subset of trials (or both).

## Definition

Teaching schedule - a function defined on an infinite set , and satisfying

for all

and

for all and .

The teaching schedule describes the way the data is disclosed: after the trial , Reality provides the lable for the object .

Weak teacher or -taught version of a confidence predictor is

,

where . In words, weak teacher is a confidence predictor whose prediction sets are based only on real lables disclosed by the end of the current trial.

An -taught (smoothed) conformal predictor is a confidence predictor that can be represented as for some (smoothed) conformal predictor .

## Examples

Ideal teacher (TCM). If and for each , then .

Slow teacher. If lag: is an increasing function, , and then is a predictor that learns the true label for each object but with a delay equal to .

Lazy teacher. If and for each , then is given the true lables immediately but not for every object.

## Validity

In case of weak teachers there is no validity in the strongest possible way (conservative validity). However, the following weaker types of validity can be defined:

• weak validity;
• strong validity;
• validity in the sense of the law of the iterated algorithm.

### Weak validity

All the statements in the section are given under the randomness assumption.

A randomized confidence predictor is asymptotically exact in probability if, for all significance levels and all probability distributions on ,

in probability,

where is the random variable defined as follows:

if ; otherwise.

Similarly, a confidence predictor is asymptotically conservative in probability if, for all significance levels and all probability distributions on ,

in probability.

Theorem Let be a teaching schedule with domain , .

• If , any -taught smoothed conformal predictor is asymptotically exact in probability.
• Otherwise, there exists an -taught smoothed conformal predictor which is not asymptotically exact in probability.

Corollary If , any -taught conformal predictor is asymptotically conservative in probability.