Spark Naive Bayes

Intro

Mahout currently has two flavors of Naive Bayes. The first is standard Multinomial Naive Bayes. The second is an implementation of Transformed Weight-normalized Complement Naive Bayes as introduced by Rennie et al. [1]. We refer to the former as Bayes and the latter as CBayes.

Where Bayes has long been a standard in text classification, CBayes is an extension of Bayes that performs particularly well on datasets with skewed classes and has been shown to be competitive with algorithms of higher complexity such as Support Vector Machines.

Implementations

The mahout math-scala library has an implemetation of both Bayes and CBayes which is further optimized in the spark module. Currently the Spark optimized version provides CLI drivers for training and testing. Mahout Spark-Naive-Bayes models can also be trained, tested and saved to the filesystem from the Mahout Spark Shell.

Preprocessing and Algorithm

As described in [1] Mahout Naive Bayes is broken down into the following steps (assignments are over all possible index values):

Let $\vec{d}=(\vec{d_1},...,\vec{d_n})$ be a set of documents; $d_{ij}$ is the count of word $i$ in document $j$.
Let $\vec{y}=(y_1,...,y_n)$ be their labels.
Let $\alpha_i$ be a smoothing parameter for all words in the vocabulary; let $\alpha=\sum_i{\alpha_i}$.
Preprocessing(via seq2Sparse) TF-IDF transformation and L2 length normalization of $\vec{d}$
1. $d_{ij} = \sqrt{d_{ij}}$
2. $d_{ij} = d_{ij}\left(\log{\frac{\sum_k1}{\sum_k\delta_{ik}+1}}+1\right)$
3. $d_{ij} =\frac{d_{ij}}{\sqrt{\sum_k{d_{kj}^2}}}$
Training: Bayes$(\vec{d},\vec{y})$ calculate term weights $w_{ci}$ as:
1. $\hat\theta_{ci}=\frac{d_{ic}+\alpha_i}{\sum_k{d_{kc}}+\alpha}$
2. $w_{ci}=\log{\hat\theta_{ci}}$
Training: CBayes$(\vec{d},\vec{y})$ calculate term weights $w_{ci}$ as:
1. $\hat\theta_{ci} = \frac{\sum_{j:y_j\neq c}d_{ij}+\alpha_i}{\sum_{j:y_j\neq c}{\sum_k{d_{kj}}}+\alpha}$
2. $w_{ci}=-\log{\hat\theta_{ci}}$
3. $w_{ci}=\frac{w_{ci}}{\sum_i \lvert w_{ci}\rvert}$
Label Assignment/Testing:
1. Let $\vec{t}= (t_1,...,t_n)$ be a test document; let $t_i$ be the count of the word $t$.
2. Label the document according to $l(t)=\arg\max_c \sum\limits_{i} t_i w_{ci}$

As we can see, the main difference between Bayes and CBayes is the weight calculation step. Where Bayes weighs terms more heavily based on the likelihood that they belong to class $c$, CBayes seeks to maximize term weights on the likelihood that they do not belong to any other class.

Running from the command line

Mahout provides CLI drivers for all above steps. Here we will give a simple overview of Mahout CLI commands used to preprocess the data, train the model and assign labels to the training set. An example script is given for the full process from data acquisition through classification of the classic 20 Newsgroups corpus.

Preprocessing: For a set of Sequence File Formatted documents in PATH_TO_SEQUENCE_FILES the mahout seq2sparse command performs the TF-IDF transformations (-wt tfidf option) and L2 length normalization (-n 2 option) as follows:
```
  $ mahout seq2sparse 
    -i ${PATH_TO_SEQUENCE_FILES} 
    -o ${PATH_TO_TFIDF_VECTORS} 
    -nv 
    -n 2
    -wt tfidf
```
Training: The model is then trained using mahout spark-trainnb. The default is to train a Bayes model. The -c option is given to train a CBayes model:
```
  $ mahout spark-trainnb
    -i ${PATH_TO_TFIDF_VECTORS} 
    -o ${PATH_TO_MODEL}
    -ow 
    -c
```
Label Assignment/Testing: Classification and testing on a holdout set can then be performed via mahout spark-testnb. Again, the -c option indicates that the model is CBayes:
```
  $ mahout spark-testnb 
    -i ${PATH_TO_TFIDF_TEST_VECTORS}
    -m ${PATH_TO_MODEL} 
    -c 
```

Command line options

Preprocessing: note: still reliant on MapReduce seq2sparse

Only relevant parameters used for Bayes/CBayes as detailed above are shown. Several other transformations can be performed by mahout seq2sparse and used as input to Bayes/CBayes. For a full list of mahout seq2Sparse options see the Creating vectors from text page.

  $ mahout seq2sparse                         
    --output (-o) output             The directory pathname for output.        
    --input (-i) input               Path to job input directory.              
    --weight (-wt) weight            The kind of weight to use. Currently TF   
                                         or TFIDF. Default: TFIDF                  
    --norm (-n) norm                 The norm to use, expressed as either a    
                                         float or "INF" if you want to use the     
                                         Infinite norm.  Must be greater or equal  
                                         to 0.  The default is not to normalize    
    --overwrite (-ow)                If set, overwrite the output directory    
    --sequentialAccessVector (-seq)  (Optional) Whether output vectors should  
                                         be SequentialAccessVectors. If set true   
                                         else false                                
    --namedVector (-nv)              (Optional) Whether output vectors should  
                                         be NamedVectors. If set true else false   

Training:

  $ mahout spark-trainnb
    --input (-i) input               Path to job input directory.                 
    --output (-o) output             The directory pathname for output.           
    --trainComplementary (-c)        Train complementary? Default is false.
    --master (-ma)                   Spark Master URL (optional). Default: "local".
                                         Note that you can specify the number of 
                                         cores to get a performance improvement, 
                                         for example "local[4]"
    --help (-h)                      Print out help                               

Testing:

  $ mahout spark-testnb   
    --input (-i) input               Path to job input directory.                  
    --model (-m) model               The path to the model built during training.   
    --testComplementary (-c)         Test complementary? Default is false.                          
    --master (-ma)                   Spark Master URL (optional). Default: "local". 
                                         Note that you can specify the number of 
                                         cores to get a performance improvement, 
                                         for example "local[4]"                        
    --help (-h)                      Print out help                                