By far the most popular algorithm for Spaced Repetition is SM2. Sites like Anki, Duolingo, Memrise, and Wanikani all chose SM2 over later SM-iterations *(eg. SM15)* because it is extremely simple, yet effective. Despite that, it still has a number of glaring issues. In this article I explain those issues, and provide a simple way to resolve them.

## Original SM2 Algorithm

The algorithm determines which items the user must review every day. To do that, it attaches the following data to every review-item:

- easiness:float – A number ≥ 1.3 representing how easy the item is, with 1.3 being the hardest. Defaults to 2.5
- consecutiveCorrectAnswers:int – How many times in a row the user has correctly answered this item
- nextDueDate:datetime – The next time this item needs to be reviewed

When the user wants to review, every item past its nextDueDate is queued. After the user reviews an item, they *(or the software)* give themselves a performanceRating on a scale from 0-5 *(0=worst, 5=best)*. Set a cutoff for an answer being “correct” *(defaults to 3)*. Then make the following changes to that item:

easiness +=

consecutiveCorrectAnswers =

nextDueDate =

## Problems with SM2

SM2 has a number of issues that limit its usefulness/effectiveness:

#### Problem: Non-generic variable ranges

The variable ranges are very specific to the original software, Supermemo. easiness is a number ≥ 1.3, while performanceRating is an integer from [0,5]

#### Solution

Normalize easiness and performanceRating to [0,1].

This requires setting a max value for easiness, which I set to 3.0. I also replaced easiness with difficulty, because it’s the more natural thing to measure.

#### Problem: Too many items per day

Because every day we do *all* the overdue items, it’s easy to encounter situations where one day you have only 5 items to review, and the next you have 50.

#### Solution

Only require doing the items that are *the most* overdue. Use “percentage” of due-date, rather than number of days, so that 3 days overdue is severe if the review cooldown was 1 day, but not severe if it was 6 months.

This allows review sessions to be small and quick. If the user has time, they can do multiple review sessions in a row. As a bonus, this allows “almost overdue” items to be reviewed, if the user has time.

#### Problem: Overdue items all treated equally

If the user completes a month-overdue item correctly, it’s likely they know it pretty well, so showing it again in 6 days is not helpful. They should get a bonus for correctly getting such overdue items correct.

Additionally, the above problem/solution allows almost-overdue items to be reviewed. These items should not be given full credit.

#### Solution

Weight the changes in due-date and difficulty by the item’s percentage overdue.

#### Other adjustments

- The quadratic term in the difficulty equation is so small it can be replaced with a simpler linear equation without adversely affecting the algorithm
- Anki, Memrise, and others prefer an initial 3 days between due-dates, instead of 6. I’ve adjusted the equations to use that preference.

## The Modified “SM2+” Algorithm

Here is the new algorithm, with all the above improvements in place.

For each review-item, store the following data:

- difficulty:float – How difficult the item is, from [0.0, 1.0]. Defaults to 0.3
*(if the software has no way of determining a better default for an item)* - daysBetweenReviews:float – How many days should occur between review attempts for this item
- dateLastReviewed:datetime – The last time this item was reviewed

When the user wants to review items, choose the top 10 (or so) items, ordered descending by percentageOverdue *(defined below)*, discarding items reviewed in the past 8 or so hours.

After an item is attempted, choose a performanceRating from [0.0, 1.0], with 1.0 being the best. Set a cutoff point for the answer being “correct” *(default is 0.6)*. Then set

percentOverdue =

difficulty +=

difficultyWeight =

daysBetweenReviews *=

## Daily Review Sessions

The above algorithm determines which items to review, but how should you handle the actual review?

That’s the topic for my next post – stay tuned!