# FSRS-6 & Spaced Repetition — Power-Law Forgetting Equations, 21 Parameters & Same-Day Memory

> Advanced mathematical specification of FSRS-6: full 21-parameter vector (w_0 through w_20), power-law decay derivation, same-day short-term stability equations, and gradient-descent optimization.

- **Canonical URL:** https://flashcards.world/flashcards/sets/d8343081-f0dc-49b3-a59b-8a750857e2f0/
- **Markdown URL:** https://flashcards.world/flashcards/sets/d8343081-f0dc-49b3-a59b-8a750857e2f0.md
- **Level:** advanced
- **Languages:** en → en
- **Total Cards:** 32

## Flashcards (32 Cards)

| # | Front (Term / Prompt) | Back (Definition / Answer) | Example | Multiple-Choice Options |
| :--- | :--- | :--- | :--- | :--- |
| 1 | Why is FSRS-6 formulated as a Markovian state-space model with state (D_t, S_t)? | Because the future memory trajectory after review t depends only on the current state (D_t, S_t), the elapsed time Δt, and the new rating G_t—requiring only two floating-point numbers per card in database storage! |  |  |
| 2 | How did FSRS-6 improve same-day memory modeling over FSRS-5 by introducing parameter w_19? | In FSRS-5, same-day stability multiplied S by exp(w_17 · (G - 3 + w_18)) regardless of how large S already was; FSRS-6 added the damping factor S^(-w_19) so mature cards reviewed twice on the same day do not experience runaway stability jumps. |  |  |
| 3 | In FSRS-6 optimization, how is 'Binary Recall Outcome' y ∈ {0, 1} defined from the 4 rating grades? | Grade 1 ('Again') maps to y = 0 (forgotten / lapse), while Grades 2, 3, and 4 ('Hard', 'Good', 'Easy') all map to y = 1 (successfully recalled). |  |  |
| 4 | What do parameters w_0, w_1, w_2, and w_3 represent in the FSRS-6 parameter vector? | The initial Stability values (in days) after a card's very first review for ratings Again (w_0), Hard (w_1), Good (w_2), and Easy (w_3) respectively: S_0(G) = w_{G-1}. |  |  |
| 5 | What roles do parameters w_8, w_9, and w_10 play in the FSRS-6 successful recall Stability formula S'_r? | w_8 sets the base exponential scaling factor (e^(w_8)), w_9 is the stability power-law decay exponent (S^(-w_9)), and w_10 governs the retrievability sensitivity term (exp(w_10 · (1 - R)) - 1). |  |  |
| 6 | How is the normalizing factor F derived from w_20 in FSRS-6 so that R(S, S) = 0.90 holds for any value of w_20? | Setting R(S, S) = (1 + F · 1)^(-w_20) = 0.90 and raising both sides to the power (-1 / w_20) yields 1 + F = 0.90^(-1 / w_20), so F = 0.90^(-1 / w_20) - 1. |  |  |
| 7 | What do parameters w_17, w_18, and w_19 control in FSRS-6? | They control same-day (short-term, t = 0 days) stability transitions: S'_short(S, G) = S · exp(w_17 · (G - 3 + w_18)) · S^(-w_19). |  |  |
| 8 | Summarize all 21 parameters of FSRS-6 by functional group. | w_0..w_3: Initial Stability S_0(1..4); w_4..w_7: Initial Difficulty & Mean-Reverting Difficulty Update; w_8..w_10: Long-Term Recall Stability Growth; w_11..w_14: Post-Lapse Stability S'_f; w_15..w_16: Hard Penalty & Easy Bonus; w_17..w_19: Same-Day Short-Term Stability; w_20: Power-Law Forgetting Curve Exponent. |  |  |
| 9 | What roles do parameters w_11, w_12, w_13, and w_14 play in the FSRS-6 post-lapse Stability formula S'_f? | When a card is forgotten (G = 1), w_11 is the base scale factor, w_12 is the difficulty penalty exponent (D^(-w_12)), w_13 is the prior-stability preservation exponent ((S + 1)^(w_13) - 1), and w_14 scales the retrievability factor (exp(w_14 · (1 - R))). |  |  |
| 10 | Confirm the value of D_0(1) (initial difficulty when rated 'Again' on the first review) from D_0(G) = w_4 - exp(w_5 · (G - 1)) + 1. | When G = 1, G - 1 = 0, so exp(0) = 1, and D_0(1) = w_4 - 1 + 1 = w_4. |  |  |
| 11 | How can FSRS-6 compute the exact probability that you will pass a 100-question exam drawn randomly from a 1,000-card deck on a future date T? | By projecting each card's elapsed time to date T, evaluating R_i(T - t_last, S_i) for all 1,000 cards, and computing the mean expected score E[Score] = (1/1000) ∑ R_i. |  |  |
| 12 | Why must the constraint 0 < w_0 ≤ w_1 ≤ w_2 ≤ w_3 hold for initial stabilities? | Because higher initial recall ratings (Again → Hard → Good → Easy) must correspond to non-decreasing initial memory stability. |  |  |
| 13 | What is parameter w_20 in FSRS-6, and why was it the headline mathematical upgrade from FSRS-4.5/5 to FSRS-6? | w_20 is the trainable power-law forgetting curve decay exponent in R(t, S) = (1 + F · t / S)^(-w_20). Earlier versions fixed this exponent (e.g., -1 in FSRS-4, -0.5 in FSRS-4.5/5), whereas FSRS-6 optimizes w_20 per user to capture individual long-term forgetting curve flatness. |  |  |
| 14 | In FSRS-4.5 and FSRS-5, where the decay exponent was fixed at w_20 = 0.5, what was the exact fractional value of F? | F = 0.90^(-1 / 0.5) - 1 = 0.90^(-2) - 1 = 100/81 - 1 = 19/81. |  |  |
| 15 | What is 'Minimum Recommended Retention (MMR)' optimization in FSRS? | A simulation-based algorithm that simulates review workloads across candidate Desired Retention values (e.g., 0.75 to 0.95) using your personal 21 parameters to find the exact R_d that minimizes (Total Review Time / Total Knowledge Acquired). |  |  |
| 16 | What do parameters w_15 and w_16 represent in FSRS-6? | w_15 is the Hard penalty multiplier (0 < w_15 < 1, applied when G = 2) and w_16 is the Easy bonus multiplier (w_16 > 1, applied when G = 4) in the successful recall stability equation. |  |  |
| 17 | Write the complete two-step Difficulty update equation in FSRS-6 for a review with grade G. | Step 1 (Linearly damped step): D_temp = D - w_6 · (G - 3) · (10 - D) / 9. Step 2 (Mean reversion): D' = clamp(w_7 · D_0(4) + (1 - w_7) · D_temp, 1, 10). |  |  |
| 18 | Write the complete FSRS-6 long-term Stability update equation S'_r(D, S, R, G) for a successful review (G ∈ {2, 3, 4}). | S'_r(D, S, R, G) = S · (1 + e^(w_8) · (11 - D) · S^(-w_9) · (e^(w_10 · (1 - R)) - 1) · h(G) · b(G)), where h(G) = w_15 if G=2 else 1, and b(G) = w_16 if G=4 else 1. |  |  |
| 19 | How does a larger vs. smaller value of w_20 change the shape of a learner's forgetting curve beyond t > S? | Wait—because F normalizes all curves to pass through (t = S, R = 0.90), a smaller w_20 produces a flatter, heavier long-term tail for t > S (slower late decay), whereas a larger w_20 approaches a steeper exponential-like decay for t > S. |  |  |
| 20 | How does the FSRS-6 optimizer train the 21 parameters (w_0 ... w_20) on a user's personal review history? | By treating the sequence of reviews on each card as a time-series state graph and minimizing Binary Cross-Entropy (Log Loss) between predicted Retrievability R(t_k, S_{k-1}) and binary recall outcomes y_k ∈ {0, 1} using mini-batch Adam / gradient descent with parameter constraints. |  |  |
| 21 | In FSRS-4, where the decay exponent was fixed at w_20 = 1.0, what was the exact fractional value of F? | F = 0.90^(-1) - 1 = 10/9 - 1 = 1/9. |  |  |
| 22 | Examine each multiplicand in S'_r: why is the term (11 - D) used? | Since D ∈ [1, 10], the factor (11 - D) ranges linearly from 10 (when D = 1, easiest) down to 1 (when D = 10, hardest), ensuring that easier cards gain up to 10× more stability increment than maximum-difficulty cards. |  |  |
| 23 | Why does the FSRS-6 optimizer apply parameter clipping / box constraints after each gradient step? | To guarantee psychological and mathematical validity—for example, ensuring w_20 ∈ [0.1, 0.8], D ∈ [1, 10], and positive stability growth parameters so the model never produces negative intervals or inverted difficulty scales. |  |  |
| 24 | How many review log entries are typically needed before FSRS-6 can run full 21-parameter optimization versus pre-training / default parameters? | Default FSRS-6 parameters (trained on hundreds of millions of global reviews) work out-of-the-box with 0 reviews; ~500–1,000 reviews enable reliable initial parameter tuning (pre-train), and ≥ 1,000+ reviews support full 21-parameter optimization. |  |  |
| 25 | What two database fields must a flashcard app store per card to run FSRS-6 natively without replaying history on every review? | Just `stability` (float, S in days) and `difficulty` (float, D ∈ [1, 10]), plus `last_review` timestamp (to compute elapsed days t). |  |  |
| 26 | What do parameters w_4 and w_5 control in FSRS-6? | They determine the initial Difficulty D_0(G) on a card's first review via D_0(G) = w_4 - exp(w_5 · (G - 1)) + 1, where w_4 is the initial difficulty when rated Again (G = 1) and w_5 controls how steeply difficulty drops for higher initial grades. |  |  |
| 27 | Prove mathematically what happens to R(t, S) = (1 + (0.9^(-1/w_20) - 1) · t / S)^(-w_20) in the limit as w_20 → ∞. | By the limit definition of the exponential function lim_{k→∞} (1 + x/k)^(-k) = e^(-x), as w_20 → ∞, the FSRS-6 power-law forgetting curve smoothly converges to the pure exponential curve R(t, S) = 0.9^(t / S)! |  |  |
| 28 | Write the complete FSRS-6 post-lapse Stability formula S'_f(D, S, R) when G = 1 (Again). | S'_f(D, S, R) = min(S, w_11 · D^(-w_12) · ((S + 1)^(w_13) - 1) · e^(w_14 · (1 - R))). |  |  |
| 29 | Why is S'_f wrapped in min(S, ...) in the FSRS-6 post-lapse stability formula? | To enforce the strict mathematical constraint that forgetting a card (G = 1) can never result in a higher Stability than the card had before the lapse. |  |  |
| 30 | In FSRS-6's same-day formula S'_short = S · exp(w_17 · (G - 3 + w_18)) · S^(-w_19), what constraint is applied when G ∈ {3, 4} (Good or Easy)? | When rated Good (3) or Easy (4) on a same-day review, the multiplier exp(w_17 · (G - 3 + w_18)) · S^(-w_19) is clamped to be ≥ 1.0 so that passing a same-day review never decreases a card's Stability. |  |  |
| 31 | How many trainable parameters does the FSRS-6 model have, and how are they indexed? | 21 trainable parameters, indexed from w_0 to w_20 (compared to 17 parameters in FSRS-4.5 and 19 parameters in FSRS-5). |  |  |
| 32 | What do parameters w_6 and w_7 control during subsequent Difficulty updates in FSRS-6? | w_6 scales the difficulty step size ΔD = -w_6 · (G - 3) based on the review rating, and w_7 is the mean-reversion weight pulling updated difficulty toward D_0(4). |  |  |

---
Study this deck interactively with FSRS-6 spaced repetition, audio review, and multiple-choice mode at [Flashcards World](https://flashcards.world/flashcards/sets/d8343081-f0dc-49b3-a59b-8a750857e2f0/).