Author: Yixuan Qiu
Date: 2023-05-05

# recosystem: Recommender System Using Parallel Matrix Factorization

#### Yixuan Qiu

#### 2023-05-05

## About recosystem Package

`recosystem` is an R wrapper of the `LIBMF` library developed by Yu-Chin Juan, Wei-Sheng Chin, Yong Zhuang, Bo-Wen Yuan, Meng-Yuan Yang, and Chih-Jen Lin (https://www.csie.ntu.edu.tw/~cjlin/libmf/), an open source library for recommender system using parallel marix factorization. (Chin, Yuan, et al. 2015)

## Highlights of LIBMF and recosystem

`LIBMF` is a high-performance C++ library for large scale matrix factorization. `LIBMF` itself is a parallelized library, meaning that users can take advantage of multicore CPUs to speed up the computation. It also utilizes some advanced CPU features to further improve the performance. (Chin, Yuan, et al. 2015)

`recosystem` is a wrapper of `LIBMF`, hence it inherits most of the features of `LIBMF`, and additionally provides a number of user-friendly R functions to simplify data processing and model building. Also, unlike most other R packages for statistical modeling that store the whole dataset and model object in memory, `LIBMF` (and hence `recosystem`) can significantly reduce memory use, for instance the constructed model that contains information for prediction can be stored in the hard disk, and output result can also be directly written into a file rather than be kept in memory.

## A Quick View of Recommender System

The main task of recommender system is to predict unknown entries in the rating matrix based on observed values, as is shown in the table below:

| item_1 | item_2 | item_3 | … | item_n
--- | --- | --- | --- | --- | ---
user_1 | 2 | 3 | ?? | … | 5
user_2 | ?? | 4 | 3 | … | ??
user_3 | 3 | 2 | ?? | … | 3
… | … | … | … | … |
user_m | 1 | ?? | 5 | … | 4

Each cell with number in it is the rating given by some user on a specific item, while those marked with question marks are unknown ratings that need to be predicted. In some other literatures, this problem may be named collaborative filtering, matrix completion, matrix recovery, etc.

A popular technique to solve the recommender system problem is the matrix factorization method. The idea is to approximate the whole rating matrix \(R_{m\times n}\) by the product of two matrices of lower dimensions, \(P_{n\times k}\) and \(Q_{n\times k}\), such that

\[R\approx PQ'\]

Let \(p_u\) be the \(u\)-th row of \(P\), and \(q_v\) be the \(v\)-th row of \(Q\), then the rating given by user \(u\) on item \(v\) would be predicted as \(p_u q'_v\).

A typical solution for \(P\) and \(Q\) is given by the following optimization problem (Chin, Zhuang, et al. 2015a, 2015b):

\[\min_{P,Q} \sum_{(u,v)\in R} \left[f(p_u,q_v;r_{u,v})+\mu_P||p_u||_1+\mu_Q||q_v||_1+\frac{\lambda_P}{2} ||p_u||_2^2+\frac{\lambda_Q}{2} ||q_v||_2^2\right]\]

where \((u,v)\) are locations of observed entries in \(R\), \(r_{u,v}\) is the observed rating, \(f\) is the loss function, and \(\mu_P,\mu_Q,\lambda_P,\lambda_Q\) are penalty parameters to avoid overfitting.

The process of solving the matrices \(P\) and \(Q\) is referred to as model training, and the selection of penalty parameters is called parameter tuning. In `recosystem`, we provide convenient functions for these two tasks, and additionally have functions for model exporting (outputing \(P\) and \(Q\) matrices) and prediction.

## Data Input and Output

Each step in the recommender system involves data input and output, as the table below shows:

Step | Input | Output
--- | --- | ---
Model training | Training data set | –
Parameter tuning | Training data set | –
Exporting model | – | User matrix \(P\), item matrix \(Q\)
Prediction | Testing data set | Predicted values

Data may have different formats and types of storage, for example the input data set may be saved in a file or stored as R objects, and users may want the output results to be directly written into file or to be returned as R objects for further processing. In `recosystem`, we use two classes, `DataSource` and `Output`, to handle data input and output in a unified way.

An object of class `DataSource` specifies the source of a data set (either training or testing), which can be created by the following two functions:

- `data_file()`: Specifies a data set from a file in the hard disk
- `data_memory()`: Specifies a data set from R objects
- `data_matrix()`: Specifies a data set from a sparse matrix

And an object of class `Output` describes how the result should be output, typically returned by the functions below:

- `out_file()`: Result should be saved to a file
- `out_memory()`: Result should be returned as R objects
- `out_nothing()`: Nothing should be output

More data source formats and output options may be supported in the future along with the development of this package.

## Data Format

The data file for training set needs to be arranged in sparse matrix triplet form, i.e., each line in the file contains three numbers

    user_index item_index rating

User index and item index may start with either 0 or 1, and this can be specified by the `index1` parameter in `data_file()` and `data_memory()`. For example, with `index1 = FALSE`, the training data file for the rating matrix in the beginning of this article may look like

    0 0 2
    0 1 3
    1 1 4
    1 2 3
    2 0 3
    2 1 2
    ...

From version 0.4 `recosystem` supports two special types of matrix factorization: the binary matrix factorization (BMF), and the one-class matrix factorization (OCMF). BMF requires ratings to take value from `{-1, 1}`, and OCMF requires all the ratings to be positive.

Testing data file is similar to training data, but since the ratings in testing data are usually unknown, the `rating` entry in testing data file can be omitted, or can be replaced by any placeholder such as `0` or `?`.

The testing data file for the same rating matrix would be

    0 2
    1 0
    2 2
    ...

Example data files are contained in the `<recosystem>/dat` (or `<recosystem>/inst/dat`, for source package) directory.

## Usage of recosystem

The usage of `recosystem` is quite simple, mainly consisting of the following steps:

1. Create a model object (a Reference Class object in R) by calling `Reco()`.
2. (Optionally) call the `$tune()` method to select best tuning parameters along a set of candidate values.
3. Train the model by calling the `$train()` method. A number of parameters can be set inside the function, possibly coming from the result of `$tune()`.
4. (Optionally) export the model via `$output()`, i.e. write the factorization matrices \(P\) and \(Q\) into files or return them as R objects.
5. Use the `$predict()` method to compute predicted values.

Below is an example on some simulated data:

    library(recosystem)
    set.seed(123) # This is a randomized algorithm
    train_set = data_file(system.file("dat", "smalltrain.txt", package = "recosystem"))
    test_set  = data_file(system.file("dat", "smalltest.txt",  package = "recosystem"))
    r = Reco()
    opts = r$tune(train_set, opts = list(dim = c(10, 20, 30), lrate = c(0.1, 0.2),
                                         costp_l1 = 0, costq_l1 = 0,
                                         nthread = 1, niter = 10))
    opts

    ## $min
    ## $min$dim
    ## [1] 20
    ##
    ## $min$costp_l1
    ## [1] 0
    ##
    ## $min$costp_l2
    ## [1] 0.1
    ##
    ## $min$costq_l1
    ## [1] 0
    ##
    ## $min$costq_l2
    ## [1] 0.01
    ##
    ## $min$lrate
    ## [1] 0.1
    ##
    ## $min$loss_fun
    ## [1] 0.9804937
    ##
    ##
    ## $res
    ##    dim costp_l1 costp_l2 costq_l1 costq_l2 lrate  loss_fun
    ## 1   10        0     0.01        0     0.01   0.1 0.9996368
    ## 2   20        0     0.01        0     0.01   0.1 1.0040111
    ## 3   30        0     0.01        0     0.01   0.1 0.9967101
    ## 4   10        0     0.10        0     0.01   0.1 0.9930384
    ## 5   20        0     0.10        0     0.01   0.1 0.9804937
    ## 6   30        0     0.10        0     0.01   0.1 0.9921565
    ## 7   10        0     0.01        0     0.10   0.1 0.9857116
    ## 8   20        0     0.01        0     0.10   0.1 1.0006225
    ## 9   30        0     0.01        0     0.10   0.1 0.9891277
    ## 10  10        0     0.10        0     0.10   0.1 0.9826748
    ## 11  20        0     0.10        0     0.10   0.1 0.9807865
    ## 12  30        0     0.10        0     0.10   0.1 0.9863404
    ## 13  10        0     0.01        0     0.01   0.2 1.1022376
    ## 14  20        0     0.01        0     0.01   0.2 1.0266608
    ## 15  30        0     0.01        0     0.01   0.2 1.0039170
    ## 16  10        0     0.10        0     0.01   0.2 1.0734307
    ## 17  20        0     0.10        0     0.01   0.2 1.0393326
    ## 18  30        0     0.10        0     0.01   0.2 1.0003177
    ## 19  10        0     0.01        0     0.10   0.2 1.0769594
    ## 20  20        0     0.01        0     0.10   0.2 1.0323938
    ## 21  30        0     0.01        0     0.10   0.2 1.0061849
    ## 22  10        0     0.10        0     0.10   0.2 1.0365456
    ## 23  20        0     0.10        0     0.10   0.2 1.0023265
    ## 24  30        0     0.10        0     0.10   0.2 1.0044131

    r$train(train_set, opts = c(opts$min, nthread = 1, niter = 20))

    ## iter      tr_rmse          obj
    ##    0       2.2673   5.3765e+04
    ##    1       1.0267   1.3667e+04
    ##    2       0.8372   1.0147e+04
    ##    3       0.7977   9.4773e+03
    ##    4       0.7703   9.0439e+03
    ##    5       0.7402   8.5967e+03
    ##    6       0.7048   8.1202e+03
    ##    7       0.6609   7.5638e+03
    ##    8       0.6133   7.0246e+03
    ##    9       0.5614   6.4770e+03
    ##   10       0.5110   5.9985e+03
    ##   11       0.4633   5.5846e+03
    ##   12       0.4203   5.2436e+03
    ##   13       0.3833   4.9761e+03
    ##   14       0.3510   4.7545e+03
    ##   15       0.3240   4.5818e+03
    ##   16       0.3005   4.4356e+03
    ##   17       0.2808   4.3158e+03
    ##   18       0.2640   4.2181e+03
    ##   19       0.2493   4.1321e+03

    ## Write predictions to file
    pred_file = tempfile()
    r$predict(test_set, out_file(pred_file))

    ## prediction output generated at /tmp/Rtmpv6QpAN/file12da161a6e28

    print(scan(pred_file, n = 10))

    ##  [1] 3.76629 2.85805 3.13870 3.22261 2.88342 2.93686 2.71680 2.96046 2.78316
    ## [10] 3.65473

    ## Or, directly return an R vector
    pred_rvec = r$predict(test_set, out_memory())
    head(pred_rvec, 10)

    ##  [1] 3.766289 2.858053 3.138700 3.222606 2.883424 2.936856 2.716800 2.960459
    ##  [9] 2.783164 3.654734

Detailed help document for each function is available in topics `?recosystem::Reco`, `?recosystem::tune`, `?recosystem::train`, `?recosystem::output` and `?recosystem::predict`.

## Performance Improvement with Extra Installation Options

To build `recosystem` from source, one needs a C++ compiler that supports the C++11 standard.

Also, there are some flags in file `src/Makevars` (`src/Makevars.win` for Windows system) that may have influential effect on performance. It is strongly suggested to set proper flags according to your type of CPU before compiling the package, in order to achieve the best performance:

1. The default `Makevars` provides generic options that should apply to most CPUs.
2. If your CPU supports SSE3 (a list of supported CPUs), add

    PKG_CPPFLAGS += -DUSESSE
    PKG_CXXFLAGS += -msse3

3. If not only SSE3 is supported but also AVX (a list of supported CPUs), add

    PKG_CPPFLAGS += -DUSEAVX
    PKG_CXXFLAGS += -mavx

After editing the `Makevars` file, run `R CMD INSTALL recosystem` on the package source directory to install `recosystem`.

## References

Chin, Wei-Sheng, Bo-Wen Yuan, Meng-Yuan Yang, Yong Zhuang, Yu-Chin Juan, and Chih-Jen Lin. 2015. “LIBMF: A Library for Parallel Matrix Factorization in Shared-Memory Systems.” https://www.csie.ntu.edu.tw/~cjlin/papers/libmf/libmf_open_source.pdf.
Chin, Wei-Sheng, Yong Zhuang, Yu-Chin Juan, and Chih-Jen Lin. 2015a. “A Fast Parallel Stochastic Gradient Method for Matrix Factorization in Shared Memory Systems.” ACM TIST. https://www.csie.ntu.edu.tw/~cjlin/papers/libmf/libmf_journal.pdf.
———. 2015b. “A Learning-Rate Schedule for Stochastic Gradient Methods to Matrix Factorization.” PAKDD. https://www.csie.ntu.edu.tw/~cjlin/papers/libmf/mf_adaptive_pakdd.pdf.