SSP-MMC-FSRS

Introduction

SSP-MMC-FSRS is an extended verson of SSP-MMC, which is an algorithm for minimizing the memorization cost in spaced repetiton. The core hypothesis of SSP-MMC is the learner will memorize a card forever when the stability exceeds a certain threshold. With this hypothesis, and the memory state-transition function (provided by FSRS), we can formulate the problem as a special case of the Markov Decision Process (MDP), i.e., a stochastic shortest path problem.

Results

• Scheduling Policy: how the intervals are calculated. When SSP-MMC is used, the intervals are chosen so that the "cost" (in minutes of studying) is minimized. When a fixed value of desired retention (DR) is used, the intervals correspond to the desired probability of recall. IVL means that constant interval lengths - in days - are used.
• Average number of reviews per day: the average number of simulated reviews of flashcards per one day of the simulated review history. Lower is better.
• Average number of minutes per day: same as above, but minutes of studying are used instead. Lower is better.
• Total knowledge at the end: the sum of probabilities of recall of all reviewed cards by the end of the simulation. It cannot be greater than the deck size. Higher is better.
• Knowledge per minute: a measure of learning efficiency. Higher is better.

Deck size = 10,000 cards, new cards per day = 10, max. reviews per day = 9,999, max. studying time per day = 360 minutes.

The best result is highlighted in bold. The worst result is highlighted in $${\color{red}red}$$.

Duration of the simulation = 365 days

Schedulling Policy	Average number of reviews per day	Average number of minutes per day	Total knowledge at the end	Knowledge per minute
SSP-MMC	54.0	16.0	3362	210
DR=0.70	31.1	14.8	$${\color{red}3053}$$	206
DR=0.73	32.7	14.4	3106	216
DR=0.76	36.1	14.7	3162	216
DR=0.79	39.3	14.8	3204	217
DR=0.82	44.6	15.4	3262	212
DR=0.85	49.3	15.4	3307	214
DR=0.88	57.2	16.3	3356	206
DR=0.91	71.1	17.7	3406	193
DR=0.94	95.0	20.6	3452	168
DR=0.97	159.2	28.4	3501	123
IVL=3	$${\color{red}603.3}$$	$${\color{red}89.2}$$	3539	$${\color{red}40}$$
IVL=7	255.7	44.9	3504	78
IVL=30	55.8	21.7	3100	143

Duration of the simulation = 3650 days

Schedulling Policy	Average number of reviews per day	Average number of minutes per day	Total knowledge at the end	Knowledge per minute
SSP-MMC	41.1	11.3	9809	867
DR=0.70	29.6	12.0	8685	724
DR=0.73	31.1	11.8	8918	756
DR=0.76	33.5	11.8	9020	762
DR=0.79	36.1	11.8	9301	788
DR=0.82	39.4	11.9	9488	800
DR=0.85	42.7	11.7	9641	824
DR=0.88	47.1	11.6	9775	841
DR=0.91	55.8	12.2	9870	809
DR=0.94	71.8	13.8	9938	722
DR=0.97	118.5	19.3	9983	516
IVL=3	$${\color{red}2383.5}$$	$${\color{red}322.8}$$	$${\color{red}8089}$$	$${\color{red}25}$$
IVL=7	1233.9	170.3	9999	59
IVL=30	286.9	48.3	9966	206

SSP-MMC performs better over longer periods of time.

Warning

Currently, the SSP-MMC matrix can be constructed for 7544 out of 9999 users (75.4%), and fails to converge for the remaining users. This indicates that the current implementation is not suitable for practical use and that further research is necessary.