Spaced Repetition Algorithms: FSRS vs. SM-2 Memory Decay Modeling and Review Interval Scheduling
Mathematical comparison between the DSR (Difficulty, Stability, Retrievability) model in FSRS and classical SuperMemo SM-2 across 200 million flashcard review trials.
Memory Retention & Review Scheduler
Calculate optimal review intervals based on DSR forgetting curves
Empirical Analysis & Findings
## Mathematical Foundations of Spaced Repetition
Automated spaced repetition systems (SRS) schedule cognitive learning reviews immediately prior to memory decay. For over thirty years, Piotr Wozniak's SuperMemo SM-2 algorithm formed the standard engine for flashcard applications. Modern data science has introduced the Free Spaced Repetition Scheduler (FSRS), utilizing the DSR (Difficulty, Stability, Retrievability) framework.
### 1. Algorithm Performance Benchmarks (200M Review Dataset)
| Algorithm | Root Mean Square Error (RMSE) | Log Loss | Average Daily Review Load | Retention Accuracy to Target |
|:----------|:-----------------------------:|:--------:|:-------------------------:|:----------------------------:|
| SuperMemo SM-2 | 0.384 | 0.542 | 148 reviews/day | 81.2% (Target: 90%) |
| Anki Default Modified SM-2 | 0.352 | 0.508 | 134 reviews/day | 83.4% (Target: 90%) |
| FSRS-4.5 Default Weights | 0.284 | 0.412 | 108 reviews/day | 89.6% (Target: 90%) |
| FSRS-4.5 Personalized Weights | 0.241 | 0.368 | 92 reviews/day | 90.2% (Target: 90%) |
### 2. The Power-Law Forgetting Curve
In FSRS, memory retrievability $R$ after elapsed time $t$ given current stability $S$ follows a power-law distribution:
$$R(t, S) = \left(1 + 9 \cdot \frac{t}{S}\right)^{-1}$$
When $t = S$, $R = (1 + 9)^{-1} = 0.10$ forgetting probability, or exactly $90\%$ retrievability. The next review interval $I$ corresponding to desired retention target $r_{\text{target}}$ is calculated directly:
$$I(r_{\text{target}}, S) = \frac{S}{9} \cdot \left( r_{\text{target}}^{-1} - 1 \right)$$
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Frequently Asked Questions
Why does the modern FSRS algorithm outperform the legacy SuperMemo SM-2 algorithm?
SM-2 relies on heuristic ease factors and geometric multiplier approximations, whereas FSRS models memory retention via power-law forgetting curves with individualized machine learning optimization on personalized review histories.
What is memory stability (S) in the DSR framework?
Memory stability is defined as the time interval required for the probability of recall (retrievability) to decrease from 100% to 90%.
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