# Paper Idea: *A Theory of Optimal Learning Efficiency in Hierarchical Knowledge Structures*

*An idea for a paper that I don't currently have the bandwidth to write.*

*Background Info:* Precision Spaced Repetition in Hierarchical Knowledge Structures

**Formalization of Learning Efficiency**

Formally introduce the idea of learning efficiency as the fraction of the time that a students spends learning new material:

```
timeSpentLearningNew = numLessons * timePerLesson
timeSpentReviewingOld = numReviews * timePerReview
learningEfficiency = timeSpentLearningNew / timeSpentReviewingOld
```

**Independent Variables**

List (and justify) the factors that influence learning efficiency:

- the encompassing density of the knowledge graph
- the spaced repetition mechanics: the interval lengths, the degree to which inaccuracy discounts a repetition, and the severity with which early repetitions are discounted
- the student's behavior -- how frequently do they complete tasks and what is their pass rate.

*Note: it is assumed that during task selection, the student is made to complete their due reviews, and the task selection method optimally compresses those due reviews into the smallest possible set of learning tasks.*

**Relationship**

For some simple graph topologies, derive a relationship in which learning efficiency is expressed as a function of those factors.