Cao et al. ICML 2010 Presented by Danushka Bollegala.
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Transcript of Cao et al. ICML 2010 Presented by Danushka Bollegala.
Transfer Learning for Collective Link Prediction in Multiple Heterogenous Domains
Cao et al. ICML 2010Presented by Danushka Bollegala.
Link Prediction
Predict links (relations) between entities Recommend items for users (MovieLens, Amazon) Recommend users for users (social recommendation) Similarity search (suggest similar web pages) Query suggestion (suggest related queries by other users)
Collective Link Prediction (CLP) Perform multiple prediction tasks for the same set of
users simultaneously▪ Predict/recommend multiple item types (books and movies)
Pros Prediction tasks might not be independent, one can
benefit from another (books vs. movies vs. food) Less affected by data sparseness (cold start problem)
Transfer Learning+Collective Link Prediction
(this paper)
Gaussian Process for
Regression (GPR)(PRML Sec. 6.4)
Link prediction = matrix factorization
Probabilistic Principal Component Analysis
(PPCA) (Bishop & Tipping, 1999)
PRML Chapter 12.
Probabilistic non-linear matrix factorization
Lawrence &Utrasun, ICML 2009
Task similarityMatrix, T
Link Modeling via NMF
Link matrix X (xi,j is the rating given by user I to item j)
Xi,j is modeled by f(ui, vj, ε) f: link function ui: latent representation of a user i
vj: latent representation of an item j ε: noise term
Generalized matrix approximation Assumption: E is Gaussian noise N(0, σ2I) Use Y = f-1(X) Then, Y follows a multivariate Gaussian
distribution.
Functions as Vectors
We can view a function as an infinite dimensional vector f(x): (f(x1), f(x2),...)T
Each point in the domain is mapped by f to a dimension in the vector
In machine learning we must find functions (e.g. linear predictors) that map input values to their corresponding output values We must also avoid over-fitting
This can be visualized as sampling from a distribution over functions with certain properties Preference bias (cf. restriction bias)
Gaussian Process (GP) (1/2)
Linear regression modelWe get different output functions y
for different weight vectors w.Let us impose a Gaussian prior over
w
Train dataset: {(x1,y1),...,(xN,yN)}Targets: y=(y1,...,yN)T
Design matrix
Gaussian Process (2/2)When we impose a Gaussian prior over
the weight vector, then the target y is also Gaussian.
K: Kernel matrix (Gram matrix)
k: kernel function
Gaussian Process: Definition Gaussian process is defined as a probability
distribution over functions y(x) such that the set of values y(x) evaluated at an arbitrary set of points x1,...,xN jointly have a Gaussian distribution.
p(x1,...,xN) is Gaussian. Often the mean is set to zero
Non-informative prior Then the kernel function fully defines the GP.
Gaussian kernel: Exponential Kernel:
Probabilistic Matrix Factorization PMF can be seen as a Gaussian Process with latent
variables (GP-LVM) [Lawrence & Utrasun ICML 2009]
Generalized matrix approximation model
Y=f-1(X) follows a multivariate Gaussian distribution
A Gaussian prior is set on U
Probabilistic PCA model byTipping & Bishop (1999)
Non-linear version
Mapping back to X
Generalized Link Functions
Each task might have a different rating distribution.
c, α, b are parameters that must be estimated from the data.
We can relax the constraint α > 0 if we have no prior knowledge regarding the negativity of the skewness of the rating distribution.
Parameter Estimation
Compute the likelihood of the dataset
Use Stochastic Gradient Descent for optimization
Non-convex optimization Sensitive to initial conditions
Experiments
Setting Use each dataset and predict multiple items
Datasets MovieLens▪ 100000 ratings, 1-5 scale ratings, 943 users, 1682
movies, 5 popular genres Book-Crossing▪ 56148 ratings, 1-10 scale, 28503 users, 9909 books, 4
most general Amazon book categories Douban▪ A social network-based recommendation serivce▪ 10000 users, 200000 items▪ Movies, books, music
Evaluation
Evaluation measure Mean Absolute Error (MAE)
Baselines I-GP: Independent Link Prediction using
GP CMF: Collective matrix factorization ▪ non GP, classical NMF
M-GP: Joint Link prediction using multi-relational GP▪ Does not consider the similarity between tasks
Proposed method = CLP-GP
My Comments
Elegant model and well-written paperFew parameters (latent space
dimension k) need to be specified All other parameters can be learnt
Applicable to a wide range of tasksCons:
Computational complexity▪ Predictions require kernel matrix inversion▪ SGD updates might not converge▪ The problem is non-convex...