CS425: Algorithms for Web Scale Data

Transcription

1 CS: Algorithms for Web Scale Data Most of the slides are from the Mining of Massive Datasets book. These slides have been modified for CS. The original slides can be accessed at:

2 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Training data 100 million ratings, 80,000 users, 17,770 movies 6 years of data: Test data Last few ratings of each user (.8 million) Evaluation criterion: Root Mean Square Error (RMSE) = 1 R σ (i,x) R r xi Ƹ r xi Netflix s system RMSE: 0.91 Competition,700+ teams $1 million prize for 10% improvement on Netflix

3 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Matrix R 17,700 movies 80,000 users

4 Matrix R 17,700 movies Training Data Set 80,000 users 1??? 1?? 1 r,6 Test Data Set True rating of user x on item i RMSE = 1 R σ (i,x) R r xi Ƹ r xi J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Predicted rating

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

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

7 Earliest and most popular collaborative filtering method Derive unknown ratings from those of similar movies (item-item variant) Define similarity measure s ij of items i and j Select k-nearest neighbors, compute the rating N(i; x): items most similar to i that were rated by x rˆ xi jn ( i; x) s ij jn ( 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: μ b x b i ^ r xi b xi baseline estimate for r xi b xi = μ + b x + b i = overall mean rating = rating deviation of user x = (avg. rating of user x) μ = (avg. rating of movie i) μ jn ( i; x) s ij ( r jn ( i; x) xj s ij Problems/Issues: 1) 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 b xj )

9 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 9 Use a weighted sum rather than weighted avg.: r xi = b xi + w ij r xj b xj A few notes: j N(i;x) N(i; x) set of movies rated by user x that are similar to movie i w ij is the interpolation weight (some real number) We allow: σ j N(i,x) w ij 1 w ij models interaction between pairs of movies (it does not depend on user x)

10 r xi = b xi + σ j N(i,x) w ij r xj b xj How to set w ij? Remember, error metric is: 1 R σ (i,x) R r xi Ƹ r xi or equivalently SSE: σ (i,x) R r xi r xi Find w ij that minimize SSE on training data! Models relationships between item i and its neighbors j w ij can be learned/estimated based on x and all other users that rated i Why is this a good idea? J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 10

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 hope the system will also predict well the unknown ratings J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 11

12 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 1 Idea: Let s set values w such 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 SSE on training data! J w = x,i b xi + j N i;x Predicted rating Think of w as a vector of numbers w ij r xj b xj r xi True rating

13 A simple way to minimize a function f(x): Compute the derivative f Start at some point y and evaluate f(y) Make a step in the reverse direction of the gradient: y = y f(y) Repeat until converged f f y + f(y) J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, y 1

14 Example: Formulation Assume we have a dataset with a single user x and items 0, 1, and. We are given all ratings, and we want to compute the weights w 01, w 0, and w 0. Rating estimate: r xi = b xi + σ j N(i,x) w ij r xj b xj Training dataset already has the correct r xi values. We will use the estimation formula to compute the unknown weights w 01, w 0, and w 0. Optimization problem: Compute w ij values to minimize: Plug in the formulas: σ (i,x) R r xi r xi minimize J w = b x0 + w 01 r x1 b x1 + w 0 r x b x r x0 + b x1 + w 01 r x0 b x0 + w 1 r x b x r x1 + b x + w 0 r x0 b x0 + w 1 r x1 b x1 r x CS Lecture 9 Mustafa Ozdal, Bilkent University 1

15 Example: Algorithm Initialize unknown variables: Iterate: w new = new w 01 new w 0 new w 1 = 0 w 01 0 w 0 0 w 1 while w new - w old > ε w old = w new w new = w old - J(w old ) is the learning rate (a parameter) How to compute J(w old )? CS Lecture 9 Mustafa Ozdal, Bilkent University 1

16 Example: Gradient-Based Update J w = b x0 + w 01 r x1 b x1 + w 0 r x b x r x0 + b x1 + w 01 r x0 b x0 + w 1 r x b x r x1 + b x + w 0 r x0 b x0 + w 1 r x1 b x1 r x J(w) = J(w) w 01 J(w) w 0 new w 01 new w 0 new w 1 = old w 01 old w 0 old w 1 η J(w) w 01 J(w) w 0 J(w) w 1 Each partial derivative is evaluated at w old. J(w) w 1 CS Lecture 9 Mustafa Ozdal, Bilkent University 16

17 Example: Computing Partial Derivatives J w = b x0 + w 01 r x1 b x1 + w 0 r x b x r x0 + b x1 + w 01 r x0 b x0 + w 1 r x b x r x1 + b x + w 0 r x0 b x0 + w 1 r x1 b x1 r x Reminder: ( ax+b ) x = ax + b a J(w) = b w x0 + w 01 r x1 b x w 0 r x b x r x0 r x1 b x1 + b x1 + w 01 r x0 b x0 + w 1 r x b x r x1 r x0 b x0 Evaluate each partial derivative at w old to compute the gradient direction. CS Lecture 9 Mustafa Ozdal, Bilkent University 17

18 We have the optimization problem, now what? Gradient descent: Iterate until convergence: w w w J where w J is the gradient (derivative evaluated on data): w J = J(w) w ij = x,i b xi + k N i;x J w = learning rate w ik r xk b xk r xi r xj b xj for j {N i; x, i, x } else J(w) = 0 w ij Note: We fix movie i, go over all r xi, for every movie j N i; x, we compute J(w) w ij while w new - w old > ε: w old = w new w new = w old - w old J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 18 x b xi + j N i;x w ij r xj b xj r xi

19 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 19 So far: r xi = b xi + σ j N(i;x) w ij r xj b xj 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

20 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 0 Global average: User average: Movie average: 1.0 Netflix: 0.91 Basic Collaborative filtering: 0.9 CF+Biases+learned weights: 0.91 Grand Prize: 0.86

21 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 1 The Color Purple Serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s 11 Lethal Weapon Geared towards males The Lion King The Princess Diaries Funny Independence Day Dumb and Dumber

22 items items J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, SVD on Netflix data: R Q P T 1 users R 1 factors Q For now let s assume we can approximate the rating matrix R as a product of thin Q P T R has 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 users P T SVD: A = U V T factors

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

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

25 items f factors items How to estimate the missing rating of user x for item i? users.? f factors 1 Q users J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, r xi = q i p x = P T f q if p xf q i = row i of Q p x = column x of P T

26 Factor J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6 The Color Purple Serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s 11 Lethal Weapon Geared Factor 1 towards males The Lion King The Princess Diaries Funny Independence Day Dumb and Dumber

27 Factor J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 7 The Color Purple Serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s 11 Lethal Weapon Geared Factor 1 towards males The Lion King The Princess Diaries Funny Independence Day Dumb and Dumber

28 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 8 n n FYI, SVD: A: Input data matrix U: Left singular vecs m A m V T V: Right singular vecs : Singular values U So in our case: SVD on Netflix data: R Q P T A = R, Q = U, P T = V T r xi = q i p x

29 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 9 We already know that SVD gives minimum reconstruction error (Sum of Squared Errors): min U,V,Σ ij A Note two things: A ij UΣV T ij SSE and RMSE are monotonically related: RMSE = 1 c SSE 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 R has missing entries!

30 items items J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, users factors SVD isn t defined when entries are missing! Use specialized methods to find P, Q min P,Q σ i,x R r xi q i p x Note: Q We don t require cols of P, Q to be orthogonal/unit length P, Q map users/movies to a latent space The most popular model among Netflix contestants users P T r xi Ƹ = q i p x factors

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32 General Concept: Overfitting Almost-linear data is fit to a linear function and a polynomial function. Polynomial model fits perfectly to data. Linear model has some error in the training set. Image source: Wikipedia CS Lecture 9 Mustafa Ozdal, Bilkent University Linear model is expected to perform better on test data, because it filters out noise.

33 items items J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Our goal is to find P and Q such that: min P,Q i,x R r xi q i p x users factors Q users P T factors

34 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 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, SSE on test data begins 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 1??? 1?? 1

35 To solve overfitting we introduce regularization: Allow rich model where there are sufficient data Shrink aggressively where data are scarce 1??? 1?? 1 min P, Q training ( r xi q i p x ) 1 x p x i q i error 1, 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,

36 Regularization min P, Q training ( r xi q i p x ) 1 x p x i q i error 1, user set regularization parameters length What happens if the user x has rated hundreds of movies? The error term will dominate, and we ll get a rich model Noise is less of an issue because we have lots of data What happens if the user x has rated only a few movies? Length term for p x will have more effect, and we ll get a simple model Same argument applies for items CS Lecture 9 Mustafa Ozdal, Bilkent University 6

37 Factor The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s 11 Lethal Weapon Factor 1 Geared towards males min P, Q training ( r xi The Princess Diaries q p ) i x min factors error + length x p x i q i The Lion King funny Independence Day J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Dumb and Dumber 7

38 Factor The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s 11 Lethal Weapon Factor 1 Geared towards males min P, Q training ( r xi The Princess Diaries q p ) i x min factors error + length x p x i q i The Lion King funny Independence Day J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Dumb and Dumber 8

39 Factor The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s 11 Lethal Weapon Factor 1 Geared towards males min P, Q training ( r xi The Princess Diaries q p ) i x min factors error + length x p x i q i The Lion King funny Independence Day J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Dumb and Dumber 9

40 Factor The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s 11 Lethal Weapon Factor 1 Geared towards males min P, Q training ( r xi The Princess Diaries q p ) i x min factors error + length x p x i q i The Lion King funny Independence Day J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, Dumb and Dumber 0

41 Want to find matrices P and Q: min ( rxi qi px ) 1 px qi P, Q training x i Gradient descent: Initialize P and Q (using SVD, pretend missing ratings are 0) Do gradient descent: P P - P Q Q - Q where Q is gradient/derivative of matrix Q: Q = [ q if ] and q if = σ x,i r xi q i p x Here q if is entry f of row q i of matrix Q Observation: Computing gradients is slow! How to compute gradient of a matrix? Compute gradient of every element independently! p xf + λ q if J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 1

42 Example min P, Q training ( r q Rewrite objective as: xi i p x 1 x x Assume we want factors per user and item: ) p i q i p x = p x0 p x1 p x q i = q i0 q i1 q i min r xi q i0 p x0 + q i1 p x1 + q i p x x,i +λ 1 σ x p x0 + p x1 + p x +λ σ i q i0 + q i1 + q i CS Lecture 9 Mustafa Ozdal, Bilkent University

43 Example min r xi q i0 p x0 + q i1 p x1 + q i p x x,i +λ 1 x p x0 + p x1 + p x p x = p x0 p x1 p x q i = q i0 q i1 q i +λ q i0 + q i1 + q i i Compute gradient for variable q i0 : q i0 = r xi (q i0 p x0 + q i1 p x1 + q i p x ) p x0 + λ q i0 x,i Do the same for every free variable CS Lecture 9 Mustafa Ozdal, Bilkent University

44 Gradient Descent - Computation Cost Q = [ q if ] and q if = σ x,i r xi q i p x p xf + λ q if How many free variables do we have? (# of users + # of items). (# of factors) Which ratings do we process to compute q if? All ratings for item i Which ratings do we process to compute p xf? All ratings for user x What is the complexity of one iteration? O(# of ratings. # of factors) CS Lecture 9 Mustafa Ozdal, Bilkent University

45 Stochastic Gradient Descent Gradient Descent (GD): Update all free variables in one step. Need to process all ratings. Stochastic Gradient Descent (SGD): Update the free variables associated with a single rating in one step. Need many more steps to converge Each step is much faster In practice: SGD much faster than GD GD: QQ σ rxi Q(r xi ) SGD: QQ μq(r xi ) CS Lecture 9 Mustafa Ozdal, Bilkent University

46 Stochastic Gradient Descent q if = r xi q i p x x,i p xf = r xi q i p x x,i p xf + λ q if q if + λ 1 p xf Which free variables are associated with rating r xi? p x = p x0 p x1.. q i = q i0 q i1.. p xk q ik CS Lecture 9 Mustafa Ozdal, Bilkent University 6

47 Stochastic Gradient Descent q if = r xi q i p x p xf + λ q if x,i p xf = r xi q i p x q if + λ 1 p xf x,i For each r xi : ε xi = (r xi q i p x ) (derivative of the error ) q i q i + μ 1 ε xi p x λ q i (update equation) p x p x + μ ε xi q i λ 1 p x (update equation) μ learning rate Note: The operations above are vector operations CS Lecture 9 Mustafa Ozdal, Bilkent University 7

48 Stochastic gradient decent: Initialize P and Q (using SVD, pretend missing ratings are 0) Then iterate over the ratings (multiple times if necessary) and update factors: For each r xi : ε xi = (r xi q i p x ) (derivative of the error ) q i q i + μ 1 ε xi p x λ q i (update equation) p x p x + μ ε xi q i λ 1 p x (update equation) for loops: For until convergence: μ learning rate For each r xi Compute gradient, do a step J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 8

49 Value of the objective function Convergence of GD vs. SGD 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, 9

50 Koren, Bell, Volinksy, IEEE Computer, 009 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 0

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52 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 = bias of movie i b i J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets,

53 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, We have expectations on the rating by user x of 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 )

54 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, r xi = μ + b x + b i + q i p x Overall mean rating Bias for user x Bias for movie i User-Movie interaction Example: Mean rating: =.7 You are a critical reviewer: your ratings are 1 star lower than the mean: b x = -1 Star Wars gets a mean rating of 0. higher than average movie: b i = + 0. Predicted rating for you on Star Wars: = =.

55 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 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 1 ), (, ) ( min

56 RMSE CF (no time bias) Basic Latent Factors Latent Factors w/ Biases Millions of parameters J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6

57 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 7 Global average: User average: Movie average: 1.0 Netflix: 0.91 Basic Collaborative filtering: 0.9 Collaborative filtering++: 0.91 Latent factors: 0.90 Latent factors+biases: 0.89 Grand Prize: 0.86

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59 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 9 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

60 Original model: r xi = +b x + b i + q i p x Add time dependence to biases: r xi = +b x (t)+ b i (t) +q i p x Make parameters b x and b i to depend on time (1) Parameterize time-dependence by linear trends () Each bin corresponds to 10 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, 60

61 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, 61

62 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6 Global average: User average: Movie average: 1.0 Netflix: 0.91 Basic Collaborative filtering: 0.9 Collaborative filtering++: 0.91 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

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

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

65 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 6 Ensemble team formed Group of other teams on leaderboard forms a new team Relies on combining their models Quickly also get a qualifying score over 10% 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

66 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 66 Submissions limited to 1 a day Only 1 final submission could be made in the last h hours before deadline BellKor team 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 BellKor submits a little early (on purpose), 0 mins before deadline Ensemble submits their final entry 0 mins later.and everyone waits.

67 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 67

68 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 68

69 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, 69 Some slides and plots borrowed from Yehuda Koren, Robert Bell and Padhraic Smyth Further reading: Y. Koren, Collaborative filtering with temporal dynamics, KDD