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!

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.

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).

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}.

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).

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.

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).

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.

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))).

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.

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.

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.

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.

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.

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).

What do parameters w_15 and w_16 represent in FSRS-6?

w_15 is the Hard penalty multiplier (0 1, applied when G = 4) in the successful recall stability equation.

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).

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.

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.

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.

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.

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.

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.

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.

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).

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.

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)!

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))).

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.

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.

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).

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).

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.

Level: advanced
Languages: en → en
Cards: 32
Markdown: FSRS-6 & Spaced Repetition — Power-Law Forgetting Equations, 21 Parameters & Same-Day Memory (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
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).