A technical explanation of FSRS
If you want a simple overview first, start here: ABC of FSRS.
In this article, I’ll explain how FSRS-4.5 works. The differences between FSRS-4.5 and FSRS-5 are outlined near the end.
Table of contents
R, Retrievability
Let’s start with the forgetting curve. In FSRS v3, an exponential function was used. In FSRS v4, the exponential function was replaced by a power function, which provided a better fit to the data. Then, in FSRS-4.5, it was replaced by a different power function which provided an even better fit. It is used in FSRS-5 as well.
The main difference between them is how fast R declines when t»S. Note that when t=S, R=90% for all three functions. This has to hold true because in FSRS, memory stability is defined as the amount of time required for R to decrease from 100% to 90%. You can play around with them here: https://www.desmos.com/calculator/au54ecrpiz.
So why does a power function provide a better fit than the exponential function if forgetting is (in theory) exponential? Let’s take two exponential curves, with S=0.2 and S=3. And then let’s take the average of the two resulting values of R. We will have 3 functions: R_1=0.9^(t/0.2), R_1=0.9^(t/3) and R=0.5*(0.9^(t/0.2)+0.9^(t/3)).
Now here’s the interesting part: if you try to approximate the resulting function (purple), the power approximation would provide a better fit than an exponential one!
Note that I displayed R^2 on the graph, but you can use any other measure to determine the goodness of fit, the conclusion will be the same.
Important takeaway number one: a superposition of two exponential functions is better approximated by a power function.
S, Stability
Now let’s take a look at the main formula of FSRS.
Here G is grade, it affects w15 and w16. R refers to retrievability at the time of the review.
Let’s simplify this formula as much as possible.
The new value of memory stability is equal to the previous value multiplied by some factor, which we will call SInc. SInc>=1, in other words, memory stability cannot decrease if the review was successful. Easy, Good and Hard all count as “success”, Again counts as a memory “lapse”. That’s why you shouldn’t use Hard as a failing grade, only as a passing grade.
SInc is equal to one plus the product of functions of three components of memory (I’ll remove the part that depends on the grade for now).
Now let’s “unfold” each of them, starting with f(D).
Important takeaway number two: the larger the value of D, the smaller the SInc value. This means that the increase in memory stability for difficult material is smaller than for easy material.
Next, let’s unfold f(S).
Important takeaway number three: the larger the value of S, the smaller the SInc value. This means that the higher the stability of the memory, the harder it becomes to make the memory even more stable. Memory stability tends to saturate.
This will likely surprise a lot of people, but the data supports it.
Finally, let’s unfold f(R).
Important takeaway number four: the smaller the value of R, the larger the SInc value. This means that the best time to review your material is when you almost forgot it (provided that you succeeded in recalling it).
If that sounds counter-intuitive, imagine if the opposite were true. Imagine that the best time to review your material is when you know it perfectly (R is very close to 100%). There is no reason to review something if you know it perfectly, so this can’t be true.
Last but not least, we need to add three more parameters: one to control the overall “scale” of the product, and two more to account for the user pressing “Hard” or “Easy”. w15 is equal to 1 if the grade is “Good” or “Easy”, and <1 if the grade is “Hard”. w16 is equal to 1 if the grade is “Hard” or “Good”, and >1 if the grade is “Easy”. In the current implementation of FSRS, 0<w15<1 and 1<w16<6.
Now all we have to do is just take the previous value of S and multiply it by SInc to obtain the new value. Hopefully that formula makes more sense to you now.
The formula for the next value of S is different if the user pressed “Again”.
min(…, S) is necessary to ensure that post-lapse stability can never be greater than stability before the lapse. w11 serves a similar purpose to e^w8 in the main formula: it just scales the whole product by some factor to provide a better fit.
An interesting detail: in the main formula, the function of D is linear: f(D)=(11-D). Here, however, f(D) is nonlinear. Me and LMSherlock have tried different formulas, and surprisingly, these provide the best fit.
There is one problem with these formulas, though. Since both formulas require the previous value of S to calculate the next value of S, they cannot be used to estimate initial stability after the first review since there is no such thing as a “zeroth review”. So initial stability has to be estimated in a completely different way.
Here’s how. First, all reviews are grouped into 4 groups based on the first grade (Again, Hard, Good, Easy). Next, intervals and outcomes of the second review are used to plot this.
On the x axis, we have t, the interval length. On the y axis, we have the proportion of cards that the user got right for that particular interval. The size of the blue circle indicates the number of reviews. A bigger circle means more reviews.
Next, we need to find a forgetting curve that provides the best fit to this data, in other words, we need to find the value of S that minimizes the difference between the measured probability of recalling a card after this many days and the predicted probability. This is done using a fast curve-fitting method, so it only takes a fraction of the overall time required to optimize parameters. I could write three or four more paragraphs about the specifics of this curve-fitting procedure, but that’s neither interesting nor very important for understanding FSRS as a whole.
The first four parameters that you see in the “FSRS parameters” window are the initial stability values.
D, Difficulty
Unlike S and R, D has no precise definition and is just a crude heuristic. Here is the formula for initial D, after the first review.
G is grade. Again=1, Hard=2, Good=3, Easy=4. We have tried turning these four values into optimizable parameters (as opposed to just using constants), but that didn’t improve accuracy.
And here is the formula for the next value of D.
There are two things going on here. First, we update D by some value which depends on the grade and is 0 if the grade is “Good”.
Next, we apply what LMSherlock calls “mean reversion”, where the current value of D is slightly reverted back to the default value, w4.
This means that if you keep pressing “Good”, difficulty will eventually converge to its default value, which is an optimizable parameter.
Putting the “mean reversion” aside, difficulty basically works like this:
Again = add a lot
Hard = add a little bit
Good = nothing
Easy = subtract a little bit
Again and Hard increase difficulty, Good doesn’t change it (again, before “mean reversion” is applied), and Easy decreases it. We’ve tried other approaches, such as “Good = add a little bit”, but nothing improved the accuracy.
The current definition of D is flawed: it doesn’t take R into account. Imagine two situations:
You pressed “Good” when the probability of recalling this card was 90.00%.
You pressed “Good” when the probability of recalling this card was 0.01%.
Clearly, these two situations are different, because in the second one it’s very surprising that you recalled a card when R was so low, whereas in the first situation it’s not surprising. In the latter case, difficulty should be adjusted by a much greater value than in the first case.
Important takeaway number five: properly defined difficulty must depend on retrievability, not only on grades.
However, a more in-depth analysis reveals that the current formula works reasonably well.
On the x axis, we have D, and on the y axis, we have predicted and measured S. Blue dots are values of memory stability that have been measured from my review history, and the orange line is the predicted value of memory stability. Of course, both are averages that require thousands of reviews to estimate accurately.
As you can see, the orange line is close to the blue dots, meaning that, on average, predicted stability is close to actual stability. Though the fit is worse for low values of D, they are also based on fewer reviews. This is based on one of my own decks. Also, I say “close”, but mean absolute percentage error (MAPE) is around 33% for my collection here, meaning that, on average, FSRS is off my 33% when predicting the value of S. Note that this depends on what you have on the X axis. For example, below is a similar graph, but for S as a function of it’s own previous value. Here, MAPE is 12%. Also, both graphs are specifically for when the user presses “Good”.
Side note: D ranges from 1 to 10, but in the built-in version of FSRS, D is displayed as a number between 0 and 1. This conversion is completely unnecessary in my opinion.
It’s important to mention that me and Sherlock have tried to incorporate R into the formulas for D, but it didn’t improve the accuracy. Even though we know that in theory D should depend on R, we don’t know how to actually add R to D in a way that is useful.
Changes in FSRS-5
1) The formula for initial D was changed.
Again=1, Hard=2, Good=3, Easy=4. This formula provides a slightly better fit. Because difficulty is clamped between 1 and 10, negative values of D are not a problem, they will simply be replaced with 1.
2) “Mean reversion” now reverses the difficulty to that of D0(4), rather than D0(3).
3) FSRS-5 updates D and S after same-day reviews, which previously were unused. Difficulty is updated using the same formula that I described in the previous section. S is updated using the following formula.
In Anki, FSRS cannot get access to the real interval lengths of same-day reviews, so it has to rely solely on grades. w17 and w18 are two new parameters.
4) Previously, the first 4 parameters - values of initial stability - were optimized without using gradient descent, using a curve-fitting method that is specific to spaced repetition. After that, they remained “frozen”. Now, after they are calculated, they can be adjusted by gradient descent, just like the other parameters. This results in a slightly better fit.
Optimization aka training
I won’t go too into detail about this, instead you can watch this video about gradient descent by 3blue1brown. The short version:
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Choose some initial values for all parameters (except the first four in our case, since they are estimated before the “main” optimization procedure).
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Change them by some small number.
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Check how much the loss function has changed. Since our case is effectively a binary classification problem (each review is either a “success” or a “lapse”), log-loss aka binary cross-entropy is used. The loss function measures the “distance” (in some mathematical sense) between predictions and real data, and the choice of the loss function depends on the task.
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Update parameters to decrease the loss.
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Keep updating the parameters based on how much it affects the loss (steps 2-4) until the loss stops decreasing, which indicates that you have reached the minimum.
Of course, it’s a lot more nuanced than that, but if you want to learn about gradient descent, there are hundreds of videos and articles on the Internet, since this is how almost every machine learning algorithm in the world is trained.