1 Introduction
Matrix factorization is a popular technique in collaborative filtering to predict the missing ratings in usermovie recommendation system. In a very recent paper of Dey et al. Dey et al. (2017)
, bayesian formulation of this problem is discussed. Hyper prameter choice was the major issue in that paper. However this problem was attempted only after suitable choice of feature dimension for user and movie feature vector. Usual method used to select such feature dimension is nothing but one dimensional grid search that select the dimension which minimizes the test error. This approach is boring as it attracts extra computational burden to select the feature dimension. In this paper we adapt a bayesian paradigm which automatically select the feature dimension along with hyper parameters which minimizes the cost function. We propose to use simulated annealing instead of straight forward metropolis hastings so that we can extract the global optimal value of the feature vector. The idea of using simulated annealing in collaborative filter is recently attempted by few researchers [
Feng and Su (2013), Shehata et al. (2017), PicotClemente et al. (2010)]. However selecting the feature dimension automatically in usermovie recommendation set up is not attempted by anyone of the author. We propose to use reversible jump MCMC in simulated annealing set up. However such optimization still requires grid search on hyper parameters. So we propose to use empirical bayes used by Atchade to take care the choice of hyper parameters. But this does not fully solve the implementation problem as such algorithm may depend on initial choice of the hyperparameters. So we modify the empirical bayes further and we call it an Adambased empirical algorithm.Bayesian model is used in many papers related to collaborative filter (Chien and George (1999), Jin and Si (2004), Xiong et al. (2010), Yu et al. (2002)). Ansari et al. (2000) propose a bayesian preference model that statistically involves several types of information useful for making recommendations, such as user preferences, user and item features and expert evaluations. A fully bayesian treatment of the Probabilistic Matrix Factorization (PMF) model is discussed by Salakhutdinov and Mnih (2008)
in which model capacity is controlled automatically under Gibbs sampler integrating over all model parameters and hyperparameters. One of the important objective of Collaborative Filter (
Su and Khoshgoftaar (2009), N. Liu et al. (2009)) is to handle the highly sparse data. Our proposed automation in the algorithm allow us not only to take care such problems, but it also makes the algorithm more scalable.The rest of the paper is orgnized as follows. In section 2, we provide usual collaborative filter algorithm and address few problems. Implementation of simulated annealing along with Reversible Jump MCMC and Empirical Bayes method is used in section 3. Results on usermovie data set is provided in Section 4. We conclude the paper in section 5.
2 Learning latent features
Definition 2.1.
:= Number of unique users,
:= Number of unique items,
:= Number of latent feature,
:= Sparse rating matrix of dimension where the element of the rating is given by user to item .
:= The user feature matrix of dimension where row represents the user feature vector .
:= The item feature matrix of dimension where row represents the item feature vector .
:= The loss function which is to be minimized.
& := User and item hyperparameters in the regularized Loss function .
:= Frobenius norm for matrices.
:= The set of useritem indices in the sparse rating matrix for which the ratings are known.
:= The set of indices of item indices for user for which the ratings are known.
:= The set of indices of user indices for item who rated it.
When latent feature dimension is known, the system minimizes the regularized loss on the set of known ratings to find optimal set of parameter vectors and , for all and .
(1) 
This Loss function is a biconvex function in and and can be iteratively optimized by regularized least square methods keeping the hyperparameters fixed.
The user and item feature matrices are first heuristically initialized using normal random matrices with iid. entries such that the product
has a mean of 3 and variance 1.
The iteration is broken into 2 steps until test loss convergence. The first step computes the regularized least squares estimate for each of the user feature vectors
from their known ratings.The second step computes the regularized least squares estimate for each of the item feature vectors from their known ratings.
The first step minimizes the below expression keeping fixed.
(2) 
The second step minimizes the below expression keeping fixed.
(3) 
The normal equations corresponding to the regularized least squares solution for user feature vectors are
(4) 
The normal equations corresponding to the regularized least squares solution for item feature vectors are
(5) 
Iteratively minimizing the loss function gives us a local minima (possibly global).
2.1 Drawbacks of the previous method
One of the major drawback of the previous method is finding the choice of and . Usual method uses grid search method. Moreover choice of feature dimension makes the grid search approach computationally less efficient.
3 Proposed Method
We propose to use simulated annealing Welsh (1989) in this set up which connects this optimization set in bayesian paradigm. Reseacher has already used this optimization in some recent work. However capturing the dimension simultaneously in simulated annealing is not available. We use reversible jump MCMC inside metropolis hastings steps of simulated annealing.
3.1 Ordinary Simulated Annealing in Usermovie set up
Given the loss function in equation1, we get the corresponding Boltzmann Distribution as posterior distribution of given ,
(6) 
where is the temperature and each component of prior vectors or user and movie feature vectors are independent and identically distributed normal variates. Variance of each of the user movie parameters are and . Clearly and are hyper parameters.
3.2 Simulated Annealing with Reversible Jump MCMC in Empirical Bayes Set up
Reversible Jump MCMC is used when dimension in the given problem itself is an unknown quantity. In this context, user and movie feature dimension is unknown. We assume maximum possible dimension as 50. At each iteration of MCMC, we choose a dimension uniformly from the set . Temparature is set in simulated annealing as . At each move, new point is chosen based on old point via an orthogonal GramChalier transformation . If and , then GramChalier orthogonal transformation uses the following relation :
Since, choice of hyperparameter and is crucial, we use empirical bayes to select them based on the data set. In one of the casestudy in usermovie set up, Dey et al. Dey et al. (2017) used empirical bayes proposed by Atchade Atchadé (2011). The algorithm is as follows :
The Stochastic approximation algorithm proposed to maximize the function of hyperparameter given the observations () as :

Generate .

Calculate
where, is the parameter vector generated from transition kernel given the hyperparameter where is such that
(7) 
Note that is the posterior function of the parameter given the data y.
First step of stochastic approximation algorithm is to draw sample from transition kernel which is our proposed reversible jump markov chain with simulated annealing algorithm to obtain a sequence of random samples from . Let and be the current iterates of the iteration sequence and be the proposal distribution. The algorithmic steps to get the next iterate is

Set : Initial choice of

,

while ,

.

Unif(0, 1).

if()

Generate Unif(0, 1) of dimension , which contains numbers from and numbers from .

; where is a Gramchalier orthogonal transformation such that

if()

accept and update the hyperparameters using empirical bayes.

else accept x and update hyperparameters using empirical bayes.


if()

Make a birth move from to . We should get so that , where is the inverse map of .

where, and

if()

accept and update the hyperparameters using empirical bayes.

else accept x and update hyperparameters using empirical bayes.

if()

Choose from dimension from .

Apply usual simulated annealing and update the hyper parameters as similar to previous one.

At each iteration decrease the temparature.
According to the algorithm, we sample one set of and using MetropolisHasting algorithm and use them to update hyperparameters in the next step.
In this case the sequence is chosen adaptively based on a wellknown algorithm called Adam.
Adam needs the following steps of tuning the hyper parameters in context of above empirical bayes set up.

The following hyper parameters are required : = step size;
Exponential decay rates for moment estimates.
: stochastic objective function with parameter . Initial choice of parameter vector. 
Assume as initial choice of parameter vector. Initialize , and .

Repeat the following steps until convergence.

[Update biased first moment e stimate]

[Update biased first moment estimate]

[Update biased first moment estimate]

[Update second raw moment estimate]

; ; [Compute biascorrected 1st moment] , [Compute biascorrected 2nd moment]


;


Return , .
4 Data Analysis and Results
The benchmark datasets used in the experiments were the 3 Movie Lens dataset. They can be found at http://grouplens.org/datasets/movielens/. We implement the above algorithm in old MovieLens small dataset which has been used with 943 users and 1682 movie. We take 80% data as training and 20% data as test set. Initial choice of hyparameters is set to and , , , , . The vectors and were initialized with iid standard normal random numbers. We stop the updates of HyperParameters and if changes in succesive two hyperparameters falls below tolerance, where tolerance is set to .
Test and Train smoothed Loss is drawn based on a fixed size window of 10 iterations in Figure1. Minimum Training Cost can be found at k=2 and Test RMSE at this point as: 1.02089331071. However the test loss stabilizes at k = 4. Converged HyperParameters: 19.439, 25.796.
5 Conclusion
The paper describes successful implementation of choice of hidden feature dimension along with automatically tuning all hyperparameters based on the data set in a collaborative filter set up. The result depends on the selection of test and trainng set. Transformation matrix calculation makes the algorithm little slow. We can modify the algorithm taking levytype proposal density in this automated simulated annealing. There some variation of the model can proposed in this direction. The work is on progress.
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 Chien and George (1999) YungHsin Chien and Edward I George. 1999. A bayesian model for collaborative filtering. In AISTATS.
 Dey et al. (2017) Arabin Kumar Dey, Raghav Somani, and Sreangsu Acharyya. 2017. A case study of empirical Bayes in a usermovie recommendation system. Communication in Statistics : Case Studies, Data Analysis and Applications 3, 1 2 (2017), 1–6.
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