Mining of Massive Datasets Jure Leskovec, AnandRajaraman, Jeff Ullman Stanford University

Transcription

1 Note to other teachers and users of these slides: We would be delighted if you found this our material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit your own needs. If you make use of a significant portion of these slides in your own lecture, please include this message, or a link to our web site: Mining of Massive Datasets Jure Leskovec, AnandRajaraman, Jeff Ullman Stanford University

2 Training data 00 million ratings, 80,000 users, 7,770 movies 6 years of data: Test data Last few ratings of each user (.8 million) Evaluation criterion:root Mean Square Error (RMSE), Netflix s system RMSE: 0.9 Competition,700+ teams $ million prize for 0% improvement on Netflix J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

3 Matrix R 7,700 movies 80,000 users J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

4 Matrix R 7,700 movies Training Data Set 80,000 users?????, Test Data Set True rating of user x on item i RMSE =, J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Predicted rating

5 The winner of the Netflix Challenge! Multi-scale modeling of the data: Combine top level, regional modeling of the data, with a refined, local view: Global: Overall deviations of users/movies Factorization: Addressing regional effects Collaborative filtering: Extract local patterns Global effects Factorization Collaborative filtering J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

6 Global: Mean movie rating:.7 stars The Sixth Senseis 0.stars above avg. Joe rates 0.stars below avg. Baseline estimation: Joewill rate The Sixth Sense stars Local neighborhood (CF/NN): Joe didn t like related movie Signs Final estimate: Joewill rate The Sixth Sense.8 stars J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6

7 Earliest and most popularcollaborative filtering method Derive unknown ratings from those of similar movies (item-item variant) Define similarity measures ij of items iand j Select k-nearest neighbors, compute the rating N(i; x):items most similar to ithat were rated by x rˆ xi = j N ( i; x) s ij j N ( i; x) s ij r xj J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, s ij similarity of items i and j r xj rating of user x on item j N(i;x) set of items similar to item i that were rated by x 7

8 In practice we get better estimates if we model deviations: ^ r xi = b xi + baseline estimate for r xi µ = overall mean rating b x = rating deviation of user x = (avg. rating of user x) µ b i = (avg. rating of movie i) µ j N ( i; x) s ij ( r j N ( i; x) xj s ij b Problems/Issues: ) Similarity measures are arbitrary )Pairwise similarities neglect interdependencies among users )Taking a weighted average can be restricting Solution:Instead of s ij use w ij that we estimate directly from data J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 8 xj )

9 Use a weighted sumrather than weighted avg.: A few notes: ; ; set of movies rated by user x that are similar to movie i is the interpolation weight (some real number) We allow:, models interaction between pairs of movies (it does not depend on user x) J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 9

10 How to set w ij?, Remember, error metric is:, or equivalently SSE:, Find w ij that minimize SSE on training data! Models relationships between item i and its neighbors j w ij can be learned/estimatedbased on xand all other users that rated i Why is this a good idea? J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 0

11 Goal: Make good recommendations Quantify goodness using RMSE: Lower RMSE better recommendations Want to make good recommendations on items that user has not yet seen. Can t really do this! Let s set build a system such that it works well on known (user, item) ratings And hopethe system will also predict well the unknown ratings J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

12 Idea:Let s set values wsuch that they work well on known (user, item) ratings How to find such values w? Idea: Define an objective function and solve the optimization problem Find w ij that minimize SSEon training data!, ; Predicted rating Think of was a vector of numbers True rating J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

13 A simple way to minimize a function : Compute the take a derivative Start at some point and evaluate Make a step in the reverse direction of the gradient: Repeat until converged J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

14 We have the optimization problem, now what? Gradient decent: Iterate until convergence: η where is the gradient (derivative evaluated on data):, ; for ;,, else Note:We fix movie i, go over all r xi, for every movie ;, we compute ; η learning rate while w new - w old > ε: w old = w new w new = w old - η w old J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

15 So far: ; Weights w ij derived based on their role; no use of an arbitrary similarity measure (w ij s ij ) Explicitly account for interrelationships among the neighboring movies Next: Latent factor model Extract regional correlations Global effects Factorization CF/NN J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

16 Global average:.96 User average:.06 Movie average:.0 Netflix: 0.9 Basic Collaborative filtering: 0.9 CF+Biases+learned weights: 0.9 Grand Prize: 0.86 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6

17 The Color Purple Serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Geared towards males The Lion King The Princess Diaries Funny Independence Day Dumb and Dumber J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 7

18 items SVD on Netflix data: R Q P T users R items factors Q For now let s assume we can approximate the rating matrix Ras a product of thin Q P T Rhas missing entries but let s ignore that for now! Basically, we will want the reconstruction error to be small on known ratings and we don t care about the values on the missing ones J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, users P T SVD: A = U Σ V T factors -.9..

19 How to estimate the missing rating of user xfor item i? items items users? factors Q factors J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, users P T q i = row i of Q p x = column x of P T

20 How to estimate the missing rating of user xfor item i? items items users? factors Q factors J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, users P T q i = row i of Q p x = column x of P T

21 How to estimate the missing rating of user xfor item i? items items users.? f factors Q f factors J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, users P T q i = row i of Q p x = column x of P T

22 The Color Purple Serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Geared Factor towards males The Lion King The Princess Diaries Factor Funny Independence Day Dumb and Dumber J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

23 The Color Purple Serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Geared Factor towards males The Lion King The Princess Diaries Factor Funny Independence Day Dumb and Dumber J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

24 n n Remember SVD: A: Input data matrix U: Left singular vecs m A m Σ V T V: Right singular vecs Σ: Singular values U So in our case: SVD on Netflix data: R Q P T A= R, Q = U, P T = ΣV T J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

25 We already know that SVD gives minimum reconstruction error (Sum of Squared Errors): min Σ,, Note two things: SSE and RMSE are monotonically related: Great news: SVD is minimizing RMSE Complication:The sum in SVD error term is over all entries (no-rating in interpreted as zero-rating). But our Rhas missing entries! J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

26 items users SVD isn t defined when entries are missing! Use specialized methods to find P, Q min, items, factors Q Note: We don t require cols of P, Qto be orthogonal/unit length P, Qmap users/movies to a latent space The most popular model among Netflix contestants users P T factors J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6

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28 Our goal is to find P and Q such tat:,, users factors items items Q users P T factors J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 8

29 Want to minimize SSE for unseen test data Idea: Minimize SSE on training data Want large k(# of factors) to capture all the signals But, SSEon testdatabegins to rise for k> This is a classical example of overfitting: With too much freedom (too many free parameters) the model starts fitting noise That is it fits too well the training data and thus not generalizing well to unseen test data????? J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 9

30 To solve overfittingwe introduce regularization: Allow rich model where there are sufficient data Shrink aggressively where data are scarce????? min P, Q training ( r xi q i p x ) + λ x p x + λ i q i error λ, λ user set regularization parameters length Note: We do not care about the raw value of the objective function, but we care in P,Q that achieve the minimum of the objective J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 0

31 The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Factor Geared towards males min P, Q training ( r q p ) xi The Princess Diaries i x + λ min factors error + λ length x p x + i q i The Lion King Factor funny Independence Day J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Dumb and Dumber

32 The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Factor Geared towards males min P, Q training ( r q p ) xi The Princess Diaries i x + λ min factors error + λ length x p x + i q i The Lion King Factor funny Independence Day J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Dumb and Dumber

33 The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Factor Geared towards males min P, Q training ( r q p ) xi The Princess Diaries i x + λ min factors error + λ length x p x + i q i The Lion King Factor funny Independence Day J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Dumb and Dumber

34 The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Factor Geared towards males min P, Q training ( r q p ) xi The Princess Diaries i x + λ min factors error + λ length x p x + i q i The Lion King Factor funny Independence Day J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Dumb and Dumber

35 Want to find matrices Pand Q: + + min ( rxi qi px ) λ px λ qi P, Q training x i Gradient decent: Initialize Pand Q (using SVD, pretend missing ratings are 0) Do gradient descent: P P-η P Q Q-η Q where Q is gradient/derivative of matrix Q: and, Here is entry fof row q i of matrix Q Observation: Computing gradients is slow! How to compute gradient of a matrix? Compute gradient of every element independently! J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

36 Gradient Descent (GD) vs. Stochastic GD Observation: where, Here is entry fof row q i of matrix Q η η, Idea:Instead of evaluating gradient over all ratings evaluate it for each individual rating and make a step GD: η SGD: Faster convergence! Need more steps but each step is computed much faster, J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6

37 Convergence of GDvs. SGD Value of the objective function Iteration/step GD improves the value of the objective function at every step. SGD improves the value but in a noisy way. GD takes fewer steps to converge but each step takes much longer to compute. In practice, SGD is much faster! J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 7

38 Stochastic gradient decent: Initialize Pand Q (using SVD, pretend missing ratings are 0) Then iterate over the ratings (multiple times if necessary) and update factors: For each r xi : for loops: For until convergence: (derivative of the error ) (update equation) (update equation) learning rate For each r xi Compute gradient, do a step J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 8

39 Koren, Bell, Volinksy, IEEE Computer, 009 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 9

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41 user bias movie bias user-movie interaction Baseline predictor Separates users and movies Benefits from insights into user s behavior Among the main practical contributions of the competition User-Movie interaction Characterizes the matching between users and movies Attracts most research in the field Benefits from algorithmic and mathematical innovations μ = overall mean rating b x = bias of user x b i = bias of movie i J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

42 We have expectations on the rating by user xof movie i, even without estimating x s attitude towards movies like i Rating scale of user x Values of other ratings user gave recently (day-specific mood, anchoring, multi-user accounts) (Recent) popularity of movie i Selection bias; related to number of ratings user gave on the same day ( frequency ) J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

43 Overall mean rating Bias for user x Bias for movie i User-Movie interaction Example: Mean rating: µ=.7 You are a critical reviewer: your ratings are star lower than the mean: b x = - Star Wars gets a mean rating of 0.higher than average movie: b i = + 0. Predicted rating for you on Star Wars: = =. J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

44 Solve: Stochastic gradient decent to find parameters Note:Both biases b x, b i as well as interactions q i, p x are treated as parameters (we estimate them) J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, regularization goodness of fit λ is selected via gridsearch on a validation set ( ) i i x x x x i i R i x x i i x xi P Q b b p q p q b b r ), (, ) ( min λ λ λ λ µ

45 CF (no time bias) Basic Latent Factors Latent Factors w/ Biases 0.9 RMSE Millions of parameters J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

46 Global average:.96 User average:.06 Movie average:.0 Netflix: 0.9 Basic Collaborative filtering: 0.9 Collaborative filtering++: 0.9 Latent factors: 0.90 Latent factors+biases: 0.89 Grand Prize: 0.86 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6

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48 Sudden rise in the average movie rating (early 00) Improvements in Netflix GUI improvements Meaning of rating changed Movie age Users prefer new movies without any reasons Older movies are just inherently better than newer ones Y. Koren, Collaborative filtering with temporal dynamics, KDD 09 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 8

49 Original model: r xi = µ +b x + b i + q i p x Add time dependence to biases: r xi = µ +b x (t)+ b i (t) +q i p x Make parameters b x and b i to depend on time () Parameterize time-dependence by linear trends () Each bin corresponds to 0 consecutive weeks Add temporal dependence to factors p x (t) user preference vector on day t Y. Koren, Collaborative filtering with temporal dynamics, KDD 09 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 9

50 RMSE CF (no time bias) Basic Latent Factors CF (time bias) Latent Factors w/ Biases + Linear time factors + Per-day user biases + CF Millions of parameters J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 0

51 Global average:.96 User average:.06 Movie average:.0 Netflix: 0.9 Basic Collaborative filtering: 0.9 Collaborative filtering++: 0.9 Latent factors: 0.90 Latent factors+biases: 0.89 Latent factors+biases+time: Still no prize! Getting desperate. Try a kitchen sink approach! Grand Prize: 0.86 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

52 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

53 June 6 th submission triggers 0-day last call J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

54 Ensemble team formed Group of other teams on leaderboardforms a new team Relies on combining their models Quickly also get a qualifying score over 0% BellKor Continue to get small improvements in their scores Realize that they are in direct competition with Ensemble Strategy Both teams carefully monitoring the leaderboard Only sure way to check for improvement is to submit a set of predictions This alerts the other team of your latest score J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

55 Submissions limited to a day Only final submission could be made in the last h hours before deadline BellKorteam member in Austria notices (by chance) that Ensemble posts a score that is slightly better than BellKor s Frantic last hours for both teams Much computer time on final optimization Carefully calibrated to end about an hour before deadline Final submissions BellKorsubmits a little early (on purpose), 0 minsbefore deadline Ensemble submits their final entry 0 mins later.and everyone waits. J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

56 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6

57 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 7

58 Some slides and plots borrowed from Yehuda Koren, Robert Bell and Padhraic Smyth Further reading: Y. Koren, Collaborative filtering with temporal dynamics, KDD J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 8