{"id":477,"date":"2016-08-02T20:34:10","date_gmt":"2016-08-03T02:34:10","guid":{"rendered":"http:\/\/www.blueraja.com\/blog\/?p=477"},"modified":"2021-05-25T16:25:16","modified_gmt":"2021-05-25T22:25:16","slug":"a-better-spaced-repetition-learning-algorithm-sm2","status":"publish","type":"post","link":"https:\/\/www.blueraja.com\/blog\/477\/a-better-spaced-repetition-learning-algorithm-sm2","title":{"rendered":"A Better Spaced Repetition Learning Algorithm: SM2+"},"content":{"rendered":"

By far the most popular algorithm for Spaced Repetition<\/a> is SM2<\/a>.\u00a0 Sites like Anki, Duolingo, Memrise, and Wanikani all chose SM2 over later SM-iterations (eg. SM15<\/a>)<\/em> because it is extremely simple, yet effective.\u00a0 Despite that, it still has a number of glaring issues.\u00a0 In this article I explain those issues, and provide a simple way to resolve them.<\/p>\n

Original SM2 Algorithm<\/h2>\n

The algorithm determines which items the user must review every day.\u00a0 To do that, it attaches the following data to every review-item:<\/p>\n

    \n
  • easiness:float<\/code> – A number\u00a0\u2265<\/span> 1.3 representing how easy the item is, with 1.3 being the hardest.\u00a0 Defaults to 2.5<\/li>\n
  • consecutiveCorrectAnswers:int<\/code> – How many times in a row the user has correctly answered this item<\/li>\n
  • nextDueDate:datetime<\/code> – The next time this item needs to be reviewed<\/li>\n<\/ul>\n

    When the user wants to review, every item past its nextDueDate<\/code> is queued.\u00a0 After the user reviews an item, they (or the software)<\/em> give themselves a performanceRating<\/code> on a scale from 0-5 (0=worst, 5=best)<\/em>.\u00a0 Set a cutoff for an answer being “correct” (defaults to 3)<\/em>. Then make the following changes to that item:<\/p>\n

    easiness<\/code> += \"-0.8 + 0.28*performanceRating + 0.02*performanceRating^2\"<\/p>\n

    consecutiveCorrectAnswers<\/code> = \"\begin{cases}<\/p>\n

    nextDueDate<\/code> = \"\begin{cases}<\/p>\n

    Problems with SM2<\/h2>\n

    SM2 has a number of issues that limit its usefulness\/effectiveness:<\/p>\n

    Problem: Non-generic variable ranges<\/h4>\n

    The variable ranges are very specific to the original software, Supermemo.\u00a0 easiness<\/code> is a number\u00a0\u2265<\/span> 1.3, while performanceRating<\/code> is an integer from [0,5]<\/p>\n

    Solution<\/h4>\n

    Normalize easiness<\/code> and performanceRating<\/code> to [0,1].<\/p>\n

    This requires setting a max value for easiness<\/code>, which I set to 3.0.\u00a0 I also replaced easiness<\/code> with difficulty<\/code>, because it’s the more natural thing to measure.<\/p>\n

    Problem: Too many items per day<\/h4>\n

    Because every day we do all<\/em> 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.<\/p>\n

    Solution<\/h4>\n

    Only require doing the items that are the most<\/em> overdue.\u00a0 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.<\/p>\n

    This allows review sessions to be small and quick.\u00a0 If the user has time, they can do multiple review sessions in a row.\u00a0 As a bonus, this allows “almost overdue” items to be reviewed, if the user has time.<\/p>\n

    Problem: Overdue items all treated equally<\/h4>\n

    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.\u00a0 They should get a bonus for correctly getting such overdue items correct.<\/p>\n

    Additionally, the above problem\/solution allows almost-overdue items to be reviewed.\u00a0 These items should not be given full credit.<\/p>\n

    Solution<\/h4>\n

    Weight the changes in due-date and difficulty by the item’s percentage overdue.<\/p>\n

    Problem: Items learned together are reviewed together<\/h4>\n

    Items that are learned together and always correctly answered will always be reviewed together at the same time, in the same order.\u00a0 This hinders learning if they’re related.<\/p>\n

    Solution<\/h4>\n

    Add a small amount of randomness to the algorithm.<\/p>\n

    Other adjustments<\/h4>\n
      \n
    • The quadratic term in the difficulty<\/code> equation is so small it can be replaced with a simpler linear equation without adversely affecting the algorithm<\/li>\n
    • Anki, Memrise, and others prefer an initial 3 days between due-dates, instead of 6.\u00a0 I’ve adjusted the equations to use that preference.<\/li>\n<\/ul>\n

      The Modified “SM2+” Algorithm<\/h2>\n

      Here is the new algorithm, with all the above improvements in place.<\/p>\n

      For each review-item, store the following data:<\/p>\n

        \n
      • difficulty:float<\/code> – How difficult the item is, from [0.0, 1.0].\u00a0 Defaults to 0.3 (if the software has no way of determining a better default for an item)<\/em><\/li>\n
      • daysBetweenReviews:float<\/code> – How many days should occur between review attempts for this item<\/li>\n
      • dateLastReviewed:datetime<\/code> – The last time this item was reviewed<\/li>\n<\/ul>\n

        When the user wants to review items, choose the top 10~20 items, ordered descending by percentageOverdue<\/code> (defined below)<\/em>, discarding items reviewed in the past 8 or so hours.<\/p>\n

        After an item is attempted, choose a performanceRating<\/code> from [0.0, 1.0], with 1.0 being the best.\u00a0 Set a cutoff point for the answer being “correct” (default is 0.6)<\/em>. Then set<\/p>\n

        percentOverdue<\/code> = \"\begin{cases}<\/p>\n

        difficulty<\/code> += \"percentOverdue*\frac{1}{17}(8-9*performanceRating)\text{, clamp to [0,1]}\"<\/p>\n

        difficultyWeight<\/code> = \"3-1.7*difficulty\"<\/p>\n

        daysBetweenReviews<\/code> *= \"\begin{cases}<\/p>\n

        Daily Review Sessions<\/h2>\n

        The above algorithm determines which items to review, but how should you handle the actual review?<\/p>\n

        That’s the topic for my next post – stay tuned!<\/p>\n","protected":false},"excerpt":{"rendered":"

        By far the most popular algorithm for Spaced Repetition is SM2.\u00a0 Sites like Anki, Duolingo, Memrise, and Wanikani all chose SM2 over later SM-iterations (eg. SM15) because it is extremely simple, yet effective.\u00a0 Despite that, it still has a number of glaring issues.\u00a0 In this article I explain those issues, and provide a simple way […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/posts\/477"}],"collection":[{"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/comments?post=477"}],"version-history":[{"count":62,"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/posts\/477\/revisions"}],"predecessor-version":[{"id":560,"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/posts\/477\/revisions\/560"}],"wp:attachment":[{"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/media?parent=477"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/categories?post=477"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blueraja.com\/blog\/wp-json\/wp\/v2\/tags?post=477"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}