# 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.*