}}
@@ -134,9 +131,9 @@ ml.describe()
-
- | user_id | movie_id | rating
-|--------|---------|-------
+
+| user_id | movie_id | rating
+|---------|----------|-------
count|2.000026e+07|2.000026e+07|2.000026e+07
mean|6.904587e+04|9.041567e+03|3.494030e+00
std|4.003863e+04|1.978948e+04|9.998490e-01
@@ -145,7 +142,7 @@ min|1.000000e+00|1.000000e+00|5.000000e-01
50%|6.914100e+04|2.167000e+03|3.500000e+00
75%|1.036370e+05|4.770000e+03|4.000000e+00
max|1.384930e+05|1.312620e+05|5.000000e+00
-
+
| user_id | movie_id | rating | time
|--------|---------|-------|-----
0|1|2|3.5|2005-04-02 23:53:47
@@ -172,7 +169,7 @@ ml[:10]
7|1|223|4.0|2005-04-02 23:46:13
8|1|253|4.0|2005-04-02 23:35:40
9|1|260|4.0|2005-04-02 23:33:46
-
+
(138494, 131263, 18179137922)
-
+
| movie_title | num_ratings | avg_rating | movie_id | | |
|------------|------------|-----------|---------|-|-|-
296|Pulp Fiction (1994)|67310.0|4.174231
@@ -270,7 +267,7 @@ movie_stats.sort_values("num_ratings", ascending=False)[:25]
608|Fargo (1996)|43272.0|4.112359
47|Seven (a.k.a. Se7en) (1995)|43249.0|4.053493
380|True Lies (1994)|43159.0|3.491149
-
+
<138494x131263 sparse matrix of type ''
with 15000197 stored elements in Compressed Sparse Column format>
-
+
| user_id | movie_id | rating | time
|--------|---------|-------|-----
13746918|94976|7371|4.5|2009-11-04 05:51:26
@@ -376,7 +373,7 @@ ml_train[:10]
15311014|105846|4226|4.5|2004-07-30 18:12:26
8514776|58812|1285|4.0|2000-04-24 20:39:46
3802643|25919|3275|2.5|2010-06-18 00:48:40
-
+
[4.5, 3.0, 3.0, 4.5, 4.0, 2.5, 5.0, 4.5, 4.0, 2.5]
-
+
[4.5, 3.0, 3.0, 4.5, 4.0, 2.5, 5.0, 4.5, 4.0, 2.5]
-
+
| movie_title | user_id | movie_id | rating | time
|------------|--------|---------|-------|-----
4229884|Jumanji (1995)|28812|2|5.0|1996-09-23 02:08:39
@@ -499,7 +496,7 @@ names.merge(ml_train[ml_train.user_id == target_user], right_on="movie_id", left
4229957|Independence Day (a.k.a. ID4) (1996)|28812|780|5.0|1996-09-23 02:09:02
4229959|Phenomenon (1996)|28812|802|5.0|1996-09-23 02:09:02
4229960|Die Hard (1988)|28812|1036|5.0|1996-09-23 02:09:02
-
+
| movie_title | movie_id |
|------------|---------|-
586|Home Alone (1990)
-
+
| movie_id_x | user_id | rating_x | rating_y
|-----------|--------|---------|---------
0|329|17593|3.0|4.0
@@ -555,7 +552,7 @@ users_df
522688|2|126271|3.0|4.0
522689|595|82760|2.0|4.0
522690|595|18306|4.5|5.0
-
+
| movie_id_x | user_id | rating_x | rating_y | rating_dev
|-----------|--------|---------|---------|-----------
0|329|17593|3.0|4.0|1.0
@@ -586,7 +583,7 @@ users_df
522688|2|126271|3.0|4.0|1.0
522689|595|82760|2.0|4.0|2.0
522690|595|18306|4.5|5.0|0.5
-
+
| movie_title | rating_dev
|------------|-----------
318|Shawshank Redemption, The (1994)|-1.391784
@@ -628,7 +625,7 @@ names.join(rating_dev, how="inner").sort_values("rating_dev")
173|Judge Dredd (1995)|0.518570
19|Ace Ventura: When Nature Calls (1995)|0.530155
160|Congo (1995)|0.559034
-
+
| user_id | movie_id | rating | rating_adj | movie_title
|--------|---------|-------|-----------|------------
4229920|28812|344|3.0|3.141987|Ace Ventura: Pet Detective (1994)
@@ -673,7 +670,7 @@ df.join(names, on="movie_id").sort_values("movie_title")
4229892|28812|50|3.0|1.683520|Usual Suspects, The (1995)
4229903|28812|208|3.0|3.250881|Waterworld (1995)
4229919|28812|339|4.0|3.727966|While You Were Sleeping (1995)
-
+
| movie_title | num_ratings
|------------|------------
802|Phenomenon (1996)|3147
@@ -754,7 +751,7 @@ names.join(num_ratings, how="inner").sort_values("num_ratings")
593|Silence of the Lambs, The (1991)|12120
480|Jurassic Park (1993)|13546
356|Forrest Gump (1994)|13847
-
+
| user_id | movie_id | rating | rating_adj | num_ratings | rating_weighted
|--------|---------|-------|-----------|------------|----------------
4229918|28812|329|4.0|3.767164|6365|23978.000326
@@ -809,7 +806,7 @@ df
4229912|28812|296|4.0|2.883755|11893|34296.500678
4229884|28812|2|5.0|4.954595|7422|36773.001211
4229953|28812|595|4.0|3.515051|9036|31761.999825
-
$$C=M^\top M$$
-
+
since $C\_{i,j}=card(S\_{j,i}(\chi))$ is the dot product of row $i$ of $M^T$ - which is column
$i$ of $M$ - and column $j$ of $M$.
@@ -943,7 +940,7 @@ We still need the other half:
$$\sum_{u \in S_{j,i}(\chi)} u_j - \sum_{u \in S_{j,i}(\chi)} u_i$$
-
+
We can apply a similar trick here. Consider first what $\sum\_{u \in
S\_{j,i}(\chi)} u\_j$ means: It is the sum of only those ratings of
@@ -961,7 +958,7 @@ $M\_j$ (consider the definition of $M\_j$) computes this, and so:
$$\sum_{u \in S_{j,i}(\chi)} u_j = M_i \cdot U_j$$
-
+
and as with $C$, since we want every pairwise dot product, this summation just
equals element $(i,j)$ of $M^\top U$. The other half of the summation,
@@ -970,13 +967,13 @@ the transpose of this matrix:
$$\sum_{u \in S_{j,i}(\chi)} u_j - \sum_{u \in S_{j,i}(\chi)} u_i = M^\top U - (M^\top U)^\top = M^\top U - U^\top M$$
-
+
So, finally, we can compute an entire deviation matrix at once like:
$$D = \left(M^\top U - (M^\top U)^\top\right) /\ M^\top M$$
-
+
where $/$ is Hadamard (i.e. elementwise) division, and $D\_{j,i} = \textrm{dev}\_{j,i}$.
@@ -990,7 +987,7 @@ Finally, the paper gives the formula to predict how user $u$ will rate movie $j$
$$
P(u)_j = \frac{1}{card(R_j)}\sum_{i\in R_j} \left(\textrm{dev}_{j,i}+u_i\right) = \frac{1}{card(R_j)}\sum_{i\in R_j} \left(D_{j,i} + U_{u,j} \right)
$$
-
+
where $R\_j = \{i | i \in S(u), i \ne j, card(S\_{j,i}(\chi)) > 0\}$, and $S(u)$ is the set of movies that user $u$ has rated. To unpack the paper's somewhat dense notation, the summation is over every movie $i$ that user $u$ rated and that at least one other user rated, except movie $j$.
@@ -998,7 +995,7 @@ We can apply the usual trick yet one more time with a little effort. The summati
$$P(u)_j = \frac{[M_u \odot (C_j > 0)] \cdot (D_j + U_u) - U_{u,j}}{M_u \cdot (C_j > 0)}$$
-
+
#### 5.2.2.4. Approximation
@@ -1006,7 +1003,7 @@ The paper also gives a formula that is a suitable approximation for larger data
$$p^{S1}(u)_j = \bar{u} + \frac{1}{card(R_j)}\sum_{i\in R_j} \textrm{dev}_{j,i}$$
-
+
where $\bar{u}$ is user $u$'s average rating. This doesn't change the formula much; we can compute $\bar{u}$ simply as column means of $U$.
@@ -1048,11 +1045,11 @@ To show that it actually gives the same result as above, and that the approximat
-
+
(4.0875210502743862, 4.0875210502743862)
-
+
Training error: MAE=0.640, RMSE=0.834
Testing error: MAE=0.657, RMSE=0.856
-
$$\hat{r}_{ui}=q_i^\top p_u$$
-
+
However, some things aren't really interactions. Some movies are just (per the ratings) overall better or worse. Some users just tend to rate everything higher or lower. We need some sort of bias built into the model to comprehend this.
@@ -1180,7 +1177,7 @@ Let's call $b_i$ the bias for movie $i$, $b_u$ the bias for user $u$, and $\mu$
$$\hat{r}_{ui}=\mu + b_i + b_u + q_i^\top p_u$$
-
+
This is the basic model we'll implement, and the same one described in the references at the top.
@@ -1190,7 +1187,7 @@ More formally, the prediction model is:
$$\hat{r}_{ui}=\mu + b_i + b_u + q_i^\top p_u$$
-
+
where:
@@ -1218,7 +1215,7 @@ $$
\frac{\partial E}{\partial b_i} &= 2 \sum_{r_{ui}} \left(\lambda b_i + r_{ui} - \hat{r}_{ui}\right)
\end{split}
$$
-
+
Gradient with respect to $p_u$ proceeds similarly:
@@ -1232,7 +1229,7 @@ p_u}q_i^\top p_u \right) + 2 \lambda p_u \\
\frac{\partial E}{\partial p_u} &= 2 \sum_{r_{ui}} \lambda p_u - \left(r_{ui} - \hat{r}_{ui}\right)q_i^\top
\end{split}
$$
-
+
Gradient with respect to $b\_u$ is identical form to $b\_i$, and gradient with respect to $q\_i$ is identical form to $p\_u$, except that the variables switch places. The full gradients then have the standard form for gradient descent, i.e. a summation of a gradient term for each individual data point, so they turn easily into update rules for each parameter (which match the ones in the Surprise link) after absorbing the leading 2 into learning rate $\gamma$ and separating out the summation over each data point. That's given below, with $e\_{ui}=r\_{ui} - \hat{r}\_{ui}$:
@@ -1245,7 +1242,7 @@ $$
\frac{\partial E}{\partial q_i} &= 2 \sum_{r_{ui}} \lambda q_i - e_{ui}p_u^\top\ \ \ &\longrightarrow q_i' &= q_i - \gamma\frac{\partial E}{\partial q_i} &= q_i + \gamma\left(e_{ui}p_u - \lambda q_i \right) \\
\end{split}
$$
-
+
The code below is a direct implementation of this by simply iteratively applying the above equations for each data point - in other words, stochastic gradient descent.
@@ -1380,7 +1377,7 @@ svd40 = SVDModel(max_movie, max_user, ml["rating"].mean(), num_factors=num_facto
svd40.train(movies_train, users_train, ratings_train, epoch_callback=at_epoch)
{{< / highlight >}}
-
+
6982/s 8928/s 10378/s 12877/s 15290/s 11574/s 13230/s
Epoch 01/20; Training: MAE=0.674 RMSE=0.874, Testing: MAE=0.677 RMSE=0.879
@@ -1423,7 +1420,7 @@ svd40.train(movies_train, users_train, ratings_train, epoch_callback=at_epoch)
52078/s 18671/s 9292/s 11493/s 12515/s 11760/s 13039/s
Epoch 20/20; Training: MAE=0.549 RMSE=0.717, Testing: MAE=0.600 RMSE=0.787
-
+
48199/s 33520/s 16937/s 13842/s 13607/s 15574/s 15431/s
Epoch 01/20; Training: MAE=0.674 RMSE=0.875, Testing: MAE=0.677 RMSE=0.878
@@ -1487,7 +1484,7 @@ svd4.train(ml_train["movie_id"].values, ml_train["user_id"].values, ml_train["ra
6090/s 11341/s 15532/s 18298/s 17158/s 14908/s 16898/s
Epoch 20/20; Training: MAE=0.599 RMSE=0.783, Testing: MAE=0.618 RMSE=0.809
-
+
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
|--|--|--|--|--|--|--|--|--|--|---|---|---|---|---|---
0||||||||||||||||
@@ -1614,7 +1611,7 @@ latent_factor_grid(svd4.q[:2,:])
13||||||||Sound of Music; Spy Kids 2: The Island of Lost...|Bring It On; Legally Blonde|Fly Away Home; Parent Trap|Sense and Sensibility; Sex and the City|||||
14|||||||Babe; Babe: Pig in the City||||Twilight|||||
15||||||||||||||||
-
+
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
|--|--|--|--|--|--|--|--|--|--|---|---|---|---|---|---
0||||||||||||||||
@@ -1651,7 +1648,7 @@ latent_factor_grid(svd4.q[2:,:])
13||||||Nightmare on Elm Street 4: The Dream Master; F...|Wes Craven's New Nightmare (Nightmare on Elm S...|Friday the 13th; Exorcist III|Candyman; Texas Chainsaw Massacre 2|Mars Attacks!; Halloween|Evil Dead II (Dead by Dawn); Re-Animator|Night of the Living Dead; Dead Alive (Braindead)||Eraserhead||
14|||||||Nightmare on Elm Street 3: Dream Warriors; Fre...|Hellbound: Hellraiser II|Nightmare on Elm Street|||||||
15|||||||Bride of Chucky (Child's Play 4)||||Texas Chainsaw Massacre|||||
-
+
| movie_title | num_ratings | avg_rating | bias | movie_id | | | |
|------------|------------|-----------|-----|---------|-|-|-|-
318|Shawshank Redemption, The (1994)|63366.0|4.446990|1.015911
@@ -1686,7 +1683,7 @@ bias.iloc[:10]
50|Usual Suspects, The (1995)|47006.0|4.334372|0.910651
102217|Bill Hicks: Revelations (1993)|50.0|3.990000|0.900622
527|Schindler's List (1993)|50054.0|4.310175|0.898633
-
+
| movie_title | num_ratings | avg_rating | bias | movie_id | | | |
|------------|------------|-----------|-----|---------|-|-|-|-
8859|SuperBabies: Baby Geniuses 2 (2004)|209.0|0.837321|-2.377202
@@ -1712,7 +1709,7 @@ bias.iloc[:-10:-1]
4775|Glitter (2001)|685.0|1.124088|-2.047287
31698|Son of the Mask (2005)|467.0|1.252677|-2.022763
5739|Faces of Death 6 (1996)|174.0|1.261494|-2.004086
-
+
| Library | Algorithm | MAE (test) | RMSE (test)
|--------|----------|-----------|------------
0||Slope One|0.656514|0.856294
@@ -1768,7 +1765,7 @@ pd.DataFrame.from_records(
2|Surprise|Random|1.144775|1.433753
3|Surprise|Slope One|0.704730|0.923331
4|Surprise|SVD|0.694890|0.900350
-