CS246: Mining Massive Datasets Jure Leskovec, Stanford University
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1 CS6: Mining Massive Datasets Jure Leskovec, Stanford University
2 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 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
3 Labels known publicly Labels only known to Netflix Training Data Held-Out Data million ratings 00 million ratings.m ratings.m ratings Quiz Set: scores posted on leaderboard Test Set: scores known only to Netflix Scores used in determining final winner //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
4 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets Matrix R 7,700 movies 80,000 users
5 Matrix R 7,700 movies Training Data Set 80,000 users????? / 0, Test Data Set True rating of user x on item i RMSE =! " //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets $ &' $ &' ) (',&). Predicted rating
6 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 6 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
7 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 7 Global: Mean movie rating:.7 stars The Sixth Sense is 0. stars above avg. Joe rates 0. stars below avg. Þ Baseline estimation: Joe will rate The Sixth Sense stars Local neighborhood (CF/NN): Joe didn t like related movie Signs Þ Final estimate: Joe will rate The Sixth Sense.8 stars
8 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 = å jîn ( i; x) å s ij jîn ( i; x) s ij r //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 8 xj 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
9 In practice we get better estimates if we model deviations: μ b x b i ^ r xi = b xi + å baseline estimate for r xi! "# = % +! " +! # = overall mean rating = rating deviation of user x = (avg. rating of user x) μ = (avg. rating of movie i) μ jîn ( i; x) å s ij ( r jîn ( i; x) ij 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 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 9 xj s - b xj )
10 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 0 Use a weighted sum rather than weighted avg.: A few notes: "% #$ = ' #$ + ) * $+ " #+ ' #+ +.($;#) (; ) set of movies rated by user x that are similar to movie i 6 is the interpolation weight (some real number) Note, we allow: (;) 6 models interaction between pairs of movies (it does not depend on user x)
11 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets! $ "# = & "# + * -(#,") ) #*! "* & "* How to set w ij? Remember, error metric is:! "#! (#,") "# or equivalently SSE: 9 (8,7) : 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?
12 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets Goal: Make good recommendations Quantify goodness using RMSE: Lower RMSE Þ 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
13 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 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! " # = % & '( + % # (* + '* & '* ',( *. (;' Predicted rating Think of w as a vector of numbers + '( 0 True rating
14 A simple way to minimize a function!(#): Compute the derivative &!(#) Start at some point ' and evaluate &!(') Make a step in the reverse direction of the gradient: ' = ' &!(') Repeat until converged * * + + -*(+) //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets +
15 We have the optimization problem, now what? Gradient decent: Iterate until convergence:!! h%! & where %! & is the gradient (derivative evaluated on data): ' ( ) = +)(-) = +- / + - / /0,/ < 6 9 /; 7 / for = A } else DE(() D( FG = H Note: We fix movie i, go over all r xi, for every movie A, we compute I&(!) ) - = / + - / /; h learning rate while w new - w old > ε: w old = w new w new = w old - h Ñw old //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets,/ < 7 / J
16 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 6 So far:! $ "# = & "# + * -(#;") ) #*! "* & "* Weights w ij derived based on their role; no use of an arbitrary similarity measure (w ij ¹ s ij ) Explicitly account for interrelationships among the neighboring movies Next: Latent factor model Extract regional correlations Global effects Factorization CF/NN
17 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 7 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
18 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 8 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
19 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 9 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 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 S V T factors -.9..
20 How to estimate the missing rating of user x for item i? items items users? factors Q factors //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets !" #$ = & $ ( # = ) & $* ( #* users P T * q i = row i of Q p x = column x of P T
21 How to estimate the missing rating of user x for item i? items items users? factors Q factors //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets !" #$ = & $ ( # = ) & $* ( #* users P T * q i = row i of Q p x = column x of P T
22 How to estimate the missing rating of user x for item i? items items users.? f factors Q f factors //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets !" #$ = & $ ( # = ) & $* ( #* users P T * q i = row i of Q p x = column x of P T
23 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets The Color Purple Serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Geared Factor towards males The Lion King The Princess Diaries Factor Funny Independence Day Dumb and Dumber
24 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets The Color Purple Serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Geared Factor towards males The Lion King The Princess Diaries Factor Funny Independence Day Dumb and Dumber
25 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets n n Remember SVD: A: Input data matrix U: Left singular vecs m A» m S V T V: Right singular vecs S: Singular values U So in our case: SVD on Netflix data: R Q P T A = R, Q = U, P T = S V T!" #$ = & $ ( #
26 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 6 We already know that SVD gives minimum reconstruction error (Sum of Squared Errors): min ( ) *+ -Σ/ 0 *+ $,&,' *+ Note two things: SSE and RMSE are monotonically related: 678 = : ; 778 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!
27 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 7 items users» items factors Q SVD isn t defined when entries are missing! Use specialized methods to find P, Q min / ( )*, *. ) $,& *,) ( )* =, *. ) Note: We don t require cols of P, Q to be orthogonal/unit length P, Q map users/movies to a latent space This was the most popular model among Netflix contestants users P T factors
28
29 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 9 Our goal is to find P and Q such tat:!"# $,& ' ( )" + " - ). ",) 0 users factors items» items Q users P T factors
30 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 0 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?????
31 To solve overfitting we introduce regularization: Allow rich model where there is sufficient data Shrink aggressively where data is scarce????? min P, Q å training ( r xi é - + ê å q + å i px ) l px l ë x i error l, l user set regularization parameters length q i ù ú û Note: We do not care about the raw value of the objective function, but we care about P,Q that achieve the minimum of the objective //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
32 The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Factor Geared towards males min P, Q å training ( r xi The Princess Diaries é - qi px ) + lêå px + å q ë x i ù ú û min factors error + l length i The Lion King Factor funny Independence Day Dumb and Dumber //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
33 The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Factor Geared towards males min P, Q å training The Princess Diaries é ( rxi - qi px ) + lêå px + å qi ë x i ù ú û min factors error + l length The Lion King Factor funny Independence Day Dumb and Dumber //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
34 The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Factor Geared towards males min P, Q å training The Princess Diaries é ( rxi - qi px ) + lêå px + å qi ë x i ù ú û min factors error + l length The Lion King Factor funny Independence Day Dumb and Dumber //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
35 The Color Purple serious Amadeus Braveheart Geared towards females Sense and Sensibility Ocean s Lethal Weapon Factor Geared towards males min P, Q å training The Princess Diaries é ( rxi - qi px ) + lêå px + å qi ë x i ù ú û min factors error + l length The Lion King Factor funny Independence Day Dumb and Dumber //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
36 Want to find matrices P and Q: é ù å - + ê å + å min ( rxi qi px ) l px l qi ú P, Q training ë x i û Gradient descent: Initialize P and Q (using SVD, pretend missing ratings are 0) Do gradient descent: P P - h ÑP Q Q - h ÑQ where ÑQ is gradient/derivative of matrix Q:!" = [!% &' ] and!% &' = -.& % & /. /.'.,& Here 6 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! + % &' //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 6
37 Gradient Descent (GD) vs. Stochastic GD Observation:!" = [!% &' ] where!% &' = * -.& % &' /.' /.'.,& Here 7 8 is entry f of row q i of matrix Q + % &' = * Ñ = hñ = h Ñ, (6 ) Idea: Instead of evaluating gradient over all ratings evaluate it for each individual rating and make a step GD: h Ñ(6 ) 6 SGD: <Ñ(6 ) Faster convergence! Need more steps but each step is computed much faster //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 7, 6
38 Convergence of GD vs. 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! //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 8
39 Stochastic gradient descent: 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 :! "# = (' "# ) # + " ) (derivative of the error ) ) # ) # + / 0! "# + " ) # (update equation) + " + " + /! "# ) # 0 + " (update equation) for loops: For until convergence: / learning rate For each r xi Compute gradient, do a step a above //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 9
40 Koren, Bell, Volinksy, IEEE Computer, 009 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 0
41
42 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 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
43 //8 Jure Leskovec, Stanford CS6: Mining 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 )
44 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets! "# = % + ( " + ( # + ) # + " 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: = =.
45 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 (and we learn them) //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets regularization goodness of fit l is selected via gridsearch on a validation set ( ) ø ö ç è æ å å å å å Î 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 l l l l µ
46 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 6 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
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48 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 8 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
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 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 9
50 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 0 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! L Getting desperate. Try a kitchen sink approach! Grand Prize: 0.86
51 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
52 June 6 th submission triggers 0-day last call //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
53 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 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 0% BellKor Continue to get small improvements in their scores Realize they are in direct competition with team Ensemble Strategy Both teams carefully monitoring the leader board Only sure way to check for improvement is to submit a set of predictions This alerts the other team of your latest score
54 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets Submissions limited to a day Only 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.
55 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets
56 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 6
57 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 7 What s the moral of the story? Submit early! J
58 //8 Jure Leskovec, Stanford CS6: Mining Massive Datasets 8 Some slides and plots borrowed from Yehuda Koren, Robert Bell and Padhraic Smyth Further reading: Y. Koren, Collaborative filtering with temporal dynamics, KDD
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