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Spaced Repetition Systems (SRS) such as Anki, SuperMemo, Mnemosyne, Memrise, etc, use their own specific algorithms for increasing your ability to remember new words and sentences. Is there evidence that certain remembering-algorithms work better than others, from a scientifically tested point of view? This would be useful to know for language learners who like to use SRS.

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A variety of spaced repetition system have been written and documented, notably on SuperMemo. For a full history, read this. This is also a fantastic resource on Spaced Repetition which explains some of its nuances. Essentially the most trusted and proven spaced repetition algorithm is SM2, developed by Wozniak, versions of which are used by Anki and Quizlet for example.

The general consensus seems to be that the more exact one tries to get with a spaced repetition algorithm, the more the returns diminish. Humans are imperfect learners, and in rare cases where perfect discipline is present, variable factors still affect the learning process. So although SuperMemo has developed a range of algorithms since SM2, SM2 is still widely favoured for its relative simplicity.

So to answer your question, yes there is limited evidence for certain remembering algorithms working better than others notably in SuperMemo's published research, but there has never been a study wide spread enough or rigorousness enough to give a satisfactory and comprehensive answer to what is the "best" algorithm is. My personal opinion is that the issues in SRS is not in discovering the most effective algorithm necessarily, but in finding a way to make it more accessible to more people. If you are asking for your personal learning, then I would stick to SM2.

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Original paper of Pimsleur, Paul (1967) "A Memory Schedule" tells about geometrical progression and 5x multiplier for repetition algorithm.

It is naive suggestion and main reason to gave geometrical progression is to overcome increased number of repetition. With geometrical increased interval progression you'll have fixed number of repetition on long run, while with linear progression you'll have infinite growth of repetition count.

Most researched and programs just use base idea without any justifications. They adjust algorithm parameters to be practically useful (like avoiding repetition of 10000 words per day or keep learners calm/non-irritated).

I was facinated by results of Jeffrey Karpicke on SRS. Check my answer for details: https://languagelearning.stackexchange.com/a/2404/2370

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