Interpretable Memory Models for Spaced Repetition

1 Introduction

• A spaced-repetition algorithm chooses when to introduce new cards and review old ones to help students memorize a deck efficiently.Students rate how well they remembered each card, and the software logs those ratings.Source: §1 · p. 1
• Table 1 is identified as an example of a review log.The introduction describes logs containing students’ ratings after card exposures.Source: Table 1 · p. 1, §1 · p. 1
• Existing scheduler data cover only their chosen review intervals, so test-set accuracy alone cannot establish whether a model will extrapolate reliably to new schedules.The paper argues that a model also needs a mechanistic interpretation to be trusted beyond the data.Source: §1 · p. 1
• SBD is substantially smaller and more interpretable than the state of the art, while retaining near state-of-the-art performance.The paper also reports simple, near-optimal solutions for common scheduling formulations given SBD.Source: §1 · p. 1

2.1 Early Systems

• The literature spans psychology, machine learning, and practical implementations, while early learning systems mostly scheduled intervals with heuristics rather than memory models or optimization.Atkinson (1972) is an exception: it models memory and plans within a study session, rather than scheduling later reviews.Source: §2 · p. 1, §2.1 · p. 1

2.2 SuperMemo

• SM-2 updates a card’s interval by multiplying it by easiness and updates easiness with a hand-picked function of the rating.Later SuperMemo work used tables of multipliers and explicitly separated scheduling from a memory model; Wo´zniak et al. (1995) proposed a system recovering SM-2 intervals at constant retention.Source: §2.2 · p. 1

2.3 Models without Schedulers

• Many memory-model studies prioritize recall prediction, using forms ranging from regressions and neural networks to cognitive models and per-card Bayesian updates.These approaches differ in functional form, and many treat scheduling as secondary or not at all.Source: §2.3 · p. 2

2.4 DHP and FSRS

• DHP pairs half-life and difficulty state variables with fitted update rules, then uses dynamic programming over a discretized state space to minimize reviews for target stability.FSRS evolved into a model-and-scheduler system; FSRS-7 adds short-term stability as a third state variable and has 34 weights.Source: §2.4 · p. 2

2.5 Benchmark and Dataset

• A public dataset of 10,000 review logs and a public leaderboard support benchmarking memory models.FSRS-7 defines the accuracy-parsimony frontier and is the closest in nature to the models presented here.Source: §2.5 · p. 2

3.1 Problem Definition

• Scheduling formalizations balance learning and effort, with effort represented as a daily review budget or a cost per review.The paper’s planning objective maximizes discounted area under the deck’s recall curve.Source: §3.1 · p. 2

3.2 Planning vs Reinforcement Learning

• Reinforcement learning learns a policy by simulating reviews, whereas planning optimizes directly over the memory model’s state.Planning imposes additional constraints on the model.Source: §3.2 · p. 2

3.3 Constraints on the Memory Model

• A useful memory model must extrapolate beyond scheduler-chosen training intervals, have simple state dynamics for planning, and update only from inputs it can itself produce.Test-set accuracy is necessary but not sufficient; these constraints leave a narrow class of models with a remaining accuracy-parsimony tradeoff.Source: §3.3 · p. 2

4 The SBD Model

• Age is compressed by an exponent, difficulty increases the forgetting-curve exponent, and recall follows a Gamma-frailty survival model.Without the repair term, the survival curve is Burr XII; Wickelgren’s power law is the special case w1 = 1.Source: §4 · p. 3
• A pass updates stability by adding the elapsed age, while a lapse shrinks it according to how surprising the failure was.Expected failures erase little stability; surprising failures erase more.Source: §4 · p. 3
• Brittleness tracks how incomplete the last review’s repair was, while difficulty tallies failures and hard passes.Brittleness is highest after failure and lowest after an easy pass.Source: §4 · p. 3

4.1 The Binary SBD Model

• The binary SBD variant is trained on binary ratings, retains seven parameters, and has self-contained state updates.The full model instead takes ratings as input without predicting them.Source: §4.1 · p. 3

4.2 Fine-Tuning

• Fine-tuning weights recent reviews more heavily and uses a Gaussian prior that pulls each weight toward its pretrained value.The prior width comes from the spread of unregularized per-user fits, and its strength falls as users accumulate reviews.Source: §4.2 · p. 3

5.1 Data and Evaluation

• The dataset contains 10,000 review logs, and users are split into training, validation, and test sets at 50%, 25%, and 25%.Models are pretrained on training users and evaluated on held-out users; fine-tuning follows the benchmark’s time-block protocol.Source: §5.1 · p. 3

5.2 Prediction Accuracy

• Fine-tuned SBD trails FSRS-7 by around 1% on BCE and AUC and 3% on RMSE(bins); pretrained SBD outperforms FSRS-7 on RMSE(bins) by 2%.Table 2 reports test-set comparisons; the benchmark variant includes same-day repeated reviews.Source: Table 2 · p. 3, §5.2 · pp. 3–4
• SBD has 80% fewer weights than FSRS-7 and a clearer mechanistic interpretation.The paper presents this parsimony alongside near state-of-the-art performance.Source: §5.2 · p. 4

6 Planning with SBD

• The card value is an infinite-horizon discounted Bellman objective, and value iteration yields the interval maximizing value at a given review cost.The objective maximizes discounted area under the deck’s recall curve while accounting for review cost.Source: §6 · p. 4
• With a budget of B reviews per day, cards compete for slots, making scheduling a restless bandit problem addressed through the Whittle relaxation.The index is the review cost at which reviewing today and waiting are equally attractive.Source: §6 · p. 4
• The resulting policy reviews the B cards with the largest indices each day and naturally handles new cards.Such index policies are asymptotically optimal under indexability and stability conditions as the deck grows with proportional budget.Source: §6 · p. 4

7 Conclusion

• SBD is considerably more parsimonious than FSRS-7 at almost no performance cost, but test-set performance alone cannot separate models in this space.The paper says accuracy gains from modifying SBD must be weighed against reduced interpretability.Source: §7 · p. 4
• Live A/B tests comparing SBD with FSRS-7 and planning with reinforcement learning are a natural next step.The paper notes that simulation cannot cleanly compare solutions end-to-end or capture user-behavior shifts in a new environment.Source: §7 · p. 4
• A/B tests between state-of-the-art algorithms are absent from the spaced-repetition literature in general.The authors identify practical reasons for this absence.Source: §7 · p. 4

• Table 3 reports pretraining weights for SBD and its binary version, together with prior widths used for fine-tuning.Source: Table 3 · p. 6
• Weights marked † are regularized in log space.Source: Table 3 · p. 6
• The prior is placed on fitted parameters; widths for w2, w4, and w5 are implied with other weights held fixed.Source: Table 3 · p. 6