Recommender Systems Jia-Bin Huang Virginia Tech Spring 2019 - - PowerPoint PPT Presentation

Recommender Systems

Jia-Bin Huang Virginia Tech

Spring 2019

ECE-5424G / CS-5824

Administrative

• HW 4 due April 10

Unsupervised Learning

• Clustering, K-Mean
• Expectation maximization
• Dimensionality reduction
• Anomaly detection
• Recommendation system

Motivating example: Monitoring machines in a data center

𝑦1 (CPU load) 𝑦2 (Memory use) 𝑦1 (CPU load) 𝑦2 (Memory use)

Multivariate Gaussian (normal) distribution

• 𝑦 ∈ 𝑆𝑜. Don’t model 𝑞 𝑦1 , 𝑞 𝑦2 , ⋯ separately
• Model 𝑞 𝑦 all in one go.
• Parameters: 𝜈 ∈ 𝑆𝑜, Σ ∈ 𝑆𝑜×𝑜 (covariance matrix)
• 𝑞 𝑦; 𝜈, Σ =

1 2𝜌 𝑜/2 Σ 1/2 exp − 𝑦 − 𝜈 ⊤Σ−1(𝑦 − 𝜈)

Multivariate Gaussian (normal) examples

Σ = 1 1 Σ = 0.6 0.6 Σ = 2 2 𝑦1 𝑦2 𝑦1 𝑦2 𝑦1 𝑦2

Multivariate Gaussian (normal) examples

Σ = 1 1 Σ = 0.6 1 Σ = 2 1 𝑦1 𝑦2 𝑦1 𝑦2 𝑦1 𝑦2

Multivariate Gaussian (normal) examples

Σ = 1 1 Σ = 1 0.5 0.5 1 Σ = 1 0.8 0.8 1 𝑦1 𝑦2 𝑦1 𝑦2 𝑦1 𝑦2

Anomaly detection using the multivariate Gaussian distribution

• 1. Fit model 𝑞 𝑦 by setting

𝜈 = 1 𝑛 ෍

𝑗=1 𝑛

𝑦(𝑗) Σ = 1 𝑛 ෍

𝑗=1 𝑛

(𝑦(𝑗)−𝜈)(𝑦(𝑗) − 𝜈)⊤ 2 Give a new example 𝑦, compute 𝑞 𝑦; 𝜈, Σ = 1 2𝜌 𝑜/2 Σ 1/2 exp − 𝑦 − 𝜈 ⊤Σ−1(𝑦 − 𝜈) Flag an anomaly if 𝑞 𝑦 < 𝜗

Original model 𝑞 𝑦1; 𝜈1, 𝜏1

2 𝑞 𝑦2; 𝜈2, 𝜏2 2 ⋯ 𝑞 𝑦𝑜; 𝜈𝑜, 𝜏𝑜 2

Manually create features to capture anomalies where 𝑦1, 𝑦2 take unusual combinations of values Computationally cheaper (alternatively, scales better) OK even if training set size is small

Original model

𝑞 𝑦; 𝜈, Σ = 1 2𝜌 𝑜/2 Σ 1/2 exp(− 𝑦 − 𝜈 ⊤Σ−1(𝑦

Recommender Systems

• Motivation
• Problem formulation
• Content-based recommendations
• Collaborative filtering
• Mean normalization

Recommender Systems

• Motivation
• Problem formulation
• Content-based recommendations
• Collaborative filtering
• Mean normalization

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Recommender Systems

• Motivation
• Problem formulation
• Content-based recommendations
• Collaborative filtering
• Mean normalization

Example: Predicting movie ratings

• User rates movies using zero to five stars

Movie Alice (1) Bob (2) Carol (3) Dave (4) Love at last

5 5

Romance forever

5 ? ?

Cute puppies of love

? 4 ?

Nonstop car chases

5 4

Swords vs. karate

5 ?

• 𝑜𝑣 = no. users
• 𝑜𝑛 = no. movies
• 𝑠 𝑗, 𝑘 = 1 if user 𝑘 has

rated movie 𝑗

• 𝑧(𝑗,𝑘) = rating given by

user 𝑘 to movie 𝑗

Recommender Systems

• Motivation
• Problem formulation
• Content-based recommendations
• Collaborative filtering
• Mean normalization

Content-based recommender systems

Movie Alice (1) Bob (2) Carol (3) Dave (4) 𝑦1 (romance) 𝑦2 (action) Love at last

5 5 0.9

Romance forever

5 ? ? 1.0 0.01

Cute puppies

• f love

? 4 ? 0.99

Nonstop car chases

5 4 0.1 1.0

Swords vs. karate

5 ? 0.9

For each user 𝑘, learn a parameter 𝜄(𝑘) ∈ 𝑆3. Predict user 𝑘 as rating movie 𝑗 with (𝜄 𝑘 )⊤𝑦(𝑗) stars.

Content-based recommender systems

Movie Alice (1) Bob (2) Carol (3) Dave (4) 𝑦1 (romance) 𝑦2 (action) Love at last

5 5 0.9

Romance forever

5 ? ? 1.0 0.01

Cute puppies

• f love

? 4 ? 0.99

Nonstop car chases

5 4 0.1 1.0

Swords vs. karate

5 ? 0.9

For each user 𝑘, learn a parameter 𝜄(𝑘) ∈ 𝑆3. Predict user 𝑘 as rating movie 𝑗 with (𝜄 𝑘 )⊤𝑦(𝑗) stars.

𝑦(3) = 1 0.99 𝜄 1 = 5 (𝜄 1 )⊤𝑦(3) = 5 ∗ 0.99 = 4.95

Problem formulation

• 𝑠 𝑗, 𝑘 = 1 if user 𝑘 has rated movie 𝑗
• 𝑧(𝑗,𝑘) = rating given by user 𝑘 to movie 𝑗
• 𝜄(𝑘) = parameter vector for user 𝑘
• 𝑦(𝑗) = feature vector for user 𝑗
• For each user 𝑘, predicted rating: (𝜄 𝑘 )⊤𝑦(𝑗)
• 𝑛(𝑘) = no. of movies rated by user j

Goal: learn 𝜄(𝑘):

min

𝜄(𝑘)

1 2𝑛(𝑘) ෍

𝑗:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 +

𝜇 2𝑛(𝑘) ෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

Optimization objective

• Learn 𝜄 𝑘 (parameter for user 𝑘):

min

𝜄(𝑘)

1 2 ෍

𝑗:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

Learn 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 : min

𝜄 1 ,𝜄 2 ,⋯,𝜄 𝑜𝑣

1 2 ෍

𝑘=1 𝑜𝑣

෍

𝑗:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑘=1 𝑜𝑣

෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

Optimization algorithm

min

𝜄(𝑘)

1 2 ෍

𝑘=1 𝑜𝑣

෍

𝑗:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑘=1 𝑜𝑣

෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

Gradient descent update:

𝜄𝑙

𝑘 ≔ 𝜄𝑙 𝑘 − 𝛽 σ𝑗:𝑠 𝑗,𝑘 =1

𝜄 𝑘

⊤ 𝑦 𝑗

− 𝑧 𝑗,𝑘 𝑦𝑙

𝑗

(for 𝑙 = 0) 𝜄𝑙

𝑘 ≔ 𝜄𝑙 𝑘 − 𝛽 σ𝑗:𝑠 𝑗,𝑘 =1( 𝜄 𝑘 ⊤ 𝑦 𝑗

− 𝑧 𝑗,𝑘 ) 𝑦𝑙

𝑗 + 𝜇 𝜄𝑙 (𝑘)

(for 𝑙 ≠ 0)

Recommender Systems

• Motivation
• Problem formulation
• Content-based recommendations
• Collaborative filtering
• Mean normalization

Problem motivation

Movie Alice (1) Bob (2) Carol (3) Dave (4) 𝑦1 (romance) 𝑦2 (action) Love at last

5 5 0.9

Romance forever

5 ? ? 1.0 0.01

Cute puppies

• f love

? 4 ? 0.99

Nonstop car chases

5 4 0.1 1.0

Swords vs. karate

5 ? 0.9

Problem motivation

Movie Alice (1) Bob (2) Carol (3) Dave (4) 𝑦1 (romance) 𝑦2 (action) Love at last

5 5 ? ?

Romance forever

5 ? ? ? ?

Cute puppies

• f love

? 4 ? ? ?

Nonstop car chases

5 4 ? ?

Swords vs. karate

5 ? ? ?

𝜄 1 = 5 𝜄 2 = 5 𝜄 3 = 5 𝜄 4 = 5 𝑦 1 = ? ? ?

Optimization algorithm

• Given 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 , to learn 𝑦(𝑗):

min

𝑦(𝑗)

1 2 ෍

𝑘:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑙=1 𝑜

𝑦𝑙

(𝑗) 2

• Given 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 , to learn 𝑦(1), 𝑦(2), ⋯ , 𝑦(𝑜𝑛):

min

𝑦(1),𝑦(2),⋯,𝑦(𝑜𝑛)

1 2 ෍

𝑗=1 𝑜𝑛

෍

𝑘:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑗=1 𝑜𝑛

෍

𝑙=1 𝑜

𝑦𝑙

(𝑗) 2

Collaborative filtering

• Given 𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛 (and movie ratings),

Can estimate 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣

• Given 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣

Can estimate 𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛

Collaborative filtering optimization

• bjective
• Given 𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛 , estimate 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣

min

𝜄 1 ,𝜄 2 ,⋯,𝜄 𝑜𝑣

1 2 ෍

𝑘=1 𝑜𝑣

෍

𝑗:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑘=1 𝑜𝑣

෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

• Given 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 , estimate 𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛

min

𝑦(1),𝑦(2),⋯,𝑦(𝑜𝑛)

1 2 ෍

𝑗=1 𝑜𝑛

෍

𝑘:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑗=1 𝑜𝑛

෍

𝑙=1 𝑜

𝑦𝑙

(𝑗) 2

Collaborative filtering optimization objective

• Given 𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛 , estimate 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣

min

𝜄 1 ,𝜄 2 ,⋯,𝜄 𝑜𝑣

1 2 ෍

𝑘=1 𝑜𝑣

෍

𝑗:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑘=1 𝑜𝑣

෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

• Given 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 , estimate 𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛

min

𝑦(1),𝑦(2),⋯,𝑦(𝑜𝑛)

1 2 ෍

𝑗=1 𝑜𝑛

෍

𝑘:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑗=1 𝑜𝑛

෍

𝑙=1 𝑜

𝑦𝑙

(𝑗) 2

• Minimize 𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛 and 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 simultaneously

𝐾 = 1 2 ෍

𝑘:𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑘=1 𝑜𝑣

෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

+ 𝜇 2 ෍

𝑗=1 𝑜𝑛

෍

𝑙=1 𝑜

𝑦𝑙

(𝑗) 2

Collaborative filtering optimization objective

𝐾(𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛 , 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 ) = 1 2 ෍

𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑘=1 𝑜𝑣

෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

+ 𝜇 2 ෍

𝑗=1 𝑜𝑛

෍

𝑙=1 𝑜

𝑦𝑙

(𝑗) 2

Collaborative filtering algorithm

• Initialize 𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛 , 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 to small random values
• Minimize 𝐾(𝑦 1 , 𝑦 2 , ⋯ , 𝑦 𝑜𝑛 , 𝜄 1 , 𝜄 2 , ⋯ , 𝜄 𝑜𝑣 ) using gradient

descent (or an advanced optimization algorithm). For every 𝑘 = 1 ⋯ 𝑜𝑣, 𝑗 = 1, ⋯ , 𝑜𝑛: 𝑦𝑙

𝑘 ≔ 𝑦𝑙 𝑘 − 𝛽

෍

𝑘:𝑠 𝑗,𝑘 =1

( 𝜄 𝑘

⊤ 𝑦 𝑗

− 𝑧 𝑗,𝑘 ) 𝜄𝑙

𝑗 + 𝜇 𝑦𝑙 (𝑗)

𝜄𝑙

𝑘 ≔ 𝜄𝑙 𝑘 − 𝛽

෍

𝑗:𝑠 𝑗,𝑘 =1

( 𝜄 𝑘

⊤ 𝑦 𝑗

− 𝑧 𝑗,𝑘 ) 𝑦𝑙

𝑗 + 𝜇 𝜄𝑙 (𝑘)

• For a user with parameter 𝜄 and movie with (learned) feature 𝑦, predict

a star rating of 𝜄⊤𝑦

Collaborative filtering

Movie Alice (1) Bob (2) Carol (3) Dave (4) Love at last

5 5

Romance forever

5 ? ?

Cute puppies of love

? 4 ?

Nonstop car chases

5 4

Swords vs. karate

5 ?

Collaborative filtering

• Predicted ratings:

𝑌 = − 𝑦 1

⊤ −

− 𝑦 2

⊤ −

⋮ − 𝑦 𝑜𝑛

⊤ −

Θ = − 𝜄 1

⊤−

− 𝜄 2

⊤−

⋮ − 𝜄 𝑜𝑣

⊤ −

Y = XΘ⊤

Low-rank matrix factorization

Finding related movies/products

• For each product 𝑗, we learn a feature vector 𝑦(𝑗) ∈ 𝑆𝑜

𝑦1: romance, 𝑦2: action, 𝑦3: comedy, …

• How to find movie 𝑘 relate to movie 𝑗?

Small 𝑦(𝑗) − 𝑦(𝑘) movie j and I are “similar”

Recommender Systems

• Motivation
• Problem formulation
• Content-based recommendations
• Collaborative filtering
• Mean normalization

Users who have not rated any movies

1 2 ෍

𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑘=1 𝑜𝑣

෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

+ 𝜇 2 ෍

𝑗=1 𝑜𝑛

෍

𝑙=1 𝑜

𝑦𝑙

(𝑗) 2

𝜄(5) = 0

Movie Alice (1) Bob (2) Carol (3) Dave (4) Eve (5) Love at last

5 5 ?

Romance forever

5 ? ? ?

Cute puppies

• f love

? 4 ? ?

Nonstop car chases

5 4 ?

Swords vs. karate

5 ? ?

Users who have not rated any movies

1 2 ෍

𝑠 𝑗,𝑘 =1

(𝜄 𝑘 )⊤ 𝑦 𝑗 − 𝑧 𝑗,𝑘

2 + 𝜇

2 ෍

𝑘=1 𝑜𝑣

෍

𝑙=1 𝑜

𝜄𝑙

𝑘 2

+ 𝜇 2 ෍

𝑗=1 𝑜𝑛

෍

𝑙=1 𝑜

𝑦𝑙

(𝑗) 2

𝜄(5) = 0

Movie Alice (1) Bob (2) Carol (3) Dave (4) Eve (5) Love at last

5 5

Romance forever

5 ? ?

Cute puppies

• f love

? 4 ?

Nonstop car chases

5 4

Swords vs. karate

5 ?

Mean normalization

For user 𝑘, on movie 𝑗 predict: 𝜄 𝑘

⊤ 𝑦(𝑗) + 𝜈𝑗

User 5 (Eve): 𝜄 5 = 0 𝜄 5

⊤ 𝑦(𝑗) + 𝜈𝑗

Learn 𝜄(𝑘), 𝑦(𝑗)

Recommender Systems

• Motivation
• Problem formulation
• Content-based recommendations
• Collaborative filtering
• Mean normalization