[BUG] [FSRS4Anki v1.6.1 Optimizer] Optimized Default Difficulty is too low on large revlog

[kuroahna]

Describe the bug
After optimizing the parameters using fsrs4anki_optimizer on a large revlog, the default difficulty is too low

let defaultDifficulty = 1.1873;
let defaultStability = 3.6988;
let difficultyDecay = -0.9949;
let stabilityDecay = -0.1412;
let retrievabilityFactor = 1.2742;
let increaseFactor = 2.9295;
let lapsesBase = -0.062;

To Reproduce
Steps to reproduce the behavior:

1. Download FSRS4Anki v1.6.1 Optimizer and import to Google Colab
2. Upload a collection with large review history (about ~300k logs). See [REMOVED]
3. Change settings to

# Here are some settings that you need to replace before running this optimizer.

filename = "collection.colpkg"
# If you upload deck file, replace it with your deck filename. E.g., ALL__Learning.apkg
# If you upload collection file, replace it with your colpgk filename. E.g., collection-2022-09-18@13-21-58.colpkg

# Replace it with your timezone. I'm in China, so I use Asia/Shanghai.
timezone = 'America/Toronto'

# Replace it with your Anki's setting in Prefernces -> Scheduling.
next_day_starts_at = 8

# Replace it if you don't want the optimizer to use the review logs before a specific date.
revlog_start_date = "2006-10-05"

3. Runtime -> Run all

Output:

Extract successfully!

revlog.csv saved!
100%|██████████| 24582/24582 [04:35<00:00, 89.13it/s] 
Trainset saved!

100%|██████████| 313064/313064 [00:49<00:00, 6335.75it/s] 
Tensorized!
train:   0%|          | 16/313064 [00:00<1:50:46, 47.10it/s]iteration: 1
f_s: [1.9995]
f_d: [4.9995]
s_w: [3.0005, -0.6995, -0.2005, 1.0005, -0.3005]
train:  10%|█         | 31323/313064 [03:43<38:44, 121.21it/s]iteration: 31307
f_s: [2.8145]
f_d: [4.2811]
s_w: [3.1621, -0.7659, -0.1151, 1.1874, -0.0596]
train:  20%|██        | 62628/313064 [07:27<33:40, 123.94it/s]iteration: 62613
f_s: [3.1593]
f_d: [3.7107]
s_w: [3.2328, -0.9031, -0.0954, 1.2932, -0.0553]
train:  30%|███       | 93934/313064 [11:09<26:09, 139.61it/s]iteration: 93919
f_s: [3.4398]
f_d: [3.2537]
s_w: [3.2627, -0.9976, -0.0962, 1.3511, -0.0815]
train:  40%|████      | 125244/313064 [14:49<21:28, 145.81it/s]iteration: 125225
f_s: [3.6134]
f_d: [2.9337]
s_w: [3.2254, -0.998, -0.1659, 1.3523, -0.0869]
train:  50%|█████     | 156550/313064 [18:27<18:03, 144.47it/s]iteration: 156531
f_s: [3.605]
f_d: [2.6452]
s_w: [3.1698, -0.9975, -0.1452, 1.339, -0.0917]
train:  60%|██████    | 187855/313064 [22:05<13:34, 153.73it/s]iteration: 187837
f_s: [3.6384]
f_d: [2.3413]
s_w: [3.1219, -0.9863, -0.0997, 1.3147, -0.0245]
train:  70%|███████   | 219161/313064 [25:38<09:30, 164.48it/s]iteration: 219143
f_s: [3.6706]
f_d: [2.0742]
s_w: [3.0706, -0.9885, -0.1018, 1.2922, -0.0602]
train:  80%|████████  | 250468/313064 [29:06<07:48, 133.49it/s]iteration: 250449
f_s: [3.7285]
f_d: [1.6587]
s_w: [3.0623, -0.9938, -0.1125, 1.317, -0.0961]
train:  90%|█████████ | 281776/313064 [32:33<03:31, 148.23it/s]iteration: 281755
f_s: [3.8183]
f_d: [1.3745]
s_w: [2.9781, -0.9904, -0.179, 1.2664, -0.0859]
train: 100%|██████████| 313064/313064 [35:58<00:00, 145.05it/s]iteration: 313061
f_s: [3.6984]
f_d: [1.1872]
s_w: [2.9299, -0.9943, -0.1408, 1.2746, -0.0626]

Training finished!

let defaultDifficulty = 1.1873;
let defaultStability = 3.6988;
let difficultyDecay = -0.9949;
let stabilityDecay = -0.1412;
let retrievabilityFactor = 1.2742;
let increaseFactor = 2.9295;
let lapsesBase = -0.062;

1:again, 2:hard, 3:good, 4:easy

first rating: 1
rating history: 1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
interval history: 0,1,2,3,5,8,13,20,30,45,65,93,132,184,252,343
difficulty history: 0,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2

first rating: 2
rating history: 2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
interval history: 0,2,4,8,14,25,43,71,115,181,279,420,621,902,1290,1817
difficulty history: 0,2.2,2.2,2.2,2.2,2.2,2.2,2.2,2.2,2.2,2.2,2.2,2.2,2.2,2.2,2.2

first rating: 3
rating history: 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
interval history: 0,4,10,25,56,119,240,460,845,1493,2551,4227,6812,10707,16451,24757
difficulty history: 0,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2

first rating: 4
rating history: 4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
interval history: 0,7,54,320,1547,6295,22147,68839,192501,491578,1160631,2559857,5319872,10492922,19763234,35730756
difficulty history: 0,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2

rating history: 3,3,3,2,3,4,3,3,1,3,3,3
interval history: 0,4,10,25,30,80,183,421,992,4,9,20,42
difficulty history: 0,1.2,1.2,1.2,1.7,1.7,0.7,0.7,0.7,1.4,1.4,1.4,1.4

Expected behavior
The default difficultly should not be as low as 1.1873

Environment

• Optimizer version: v1.6.1

Additional context
I believe the issue has come from the commit 4185f3d

Change the learning rate from lr=1e-4 to lr=5e-4 might be too aggressive?

[L-M-Sherlock]

But the default difficulty deceased monotonically in your case. I will test your collection tomorrow.

[kuroahna]

Hm actually, you might be right. A large majority of my cards have ~250% ease

And looking at the interval history if I keep pressing Good:

first rating: 3
rating history: 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
interval history: 0,4,10,25,56,119,240,460,845,1493,2551,4227,6812,10707,16451,24757
difficulty history: 0,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2,1.2

The change between each interval is:

0 -> 4 = N/A
4 -> 10 = 10 / 4 = 2.5 = 250%
10 -> 25 = 25 / 10 = 2.5 = 250%
25 -> 56 = 56/ 25 = 2.24 = 224%
56 -> 119 = 119 / 56 = 2.125 = 212.5%
119 -> 240 = 240 / 119 = 2.017 = 201.7%
...

This seems correct since most of my cards have 250% and I have about ~88% retention rate for my young cards. The FSRS4anki algorithm starts with 250% change for the interval and then it slowly decreases the change in interval as I keep pressing good.

In version FSRS4Anki v1.4.3 Optimizer however, I got the results:

let defaultDifficulty = 3.835;
let defaultStability = 3.2191;
let difficultyDecay = -0.9131;
let stabilityDecay = -0.0872;
let retrievabilityFactor = 1.2571;
let increaseFactor = 3.2025;
let lapsesBase = -0.0493;

stability, difficulty, lapses
     3.22        3.83       0
     5.77        3.84       0
    10.62        3.84       0
    19.06        3.83       0
    32.91        3.83       0
    55.86        3.83       0
    93.04        3.83       0
   152.10        3.83       0
   244.58        3.83       0
   387.58        3.83       0
   605.15        3.83       0
   931.47        3.83       0
  1415.08        3.83       0
  2123.76        3.83       0
  3150.51        3.83       0
0,3,6,11,19,33,56,93,152,245,388,605,931,1415,2124,3151

The change between each interval here is:

0 -> 3 = N/A
3 -> 6 = 6 / 3 = 2 = 200%
6 -> 11 = 11 /  6 = 1.83 = 183%
11 -> 19 = 19 / 11 = 1.73 = 173%
19 -> 33 = 33 / 19 = 1.74 = 174%
33 -> 56 = 56 / 33 = 1.70 = 170%
...

With these parameters, it starts with a change in interval of 200%, not 250%. It does seem like these new parameters are correct. I guess the larger learning rate allowed it to converge faster.