Contents

Training Feature Generation
CRF Training Function
Testing Feature Generation
Inference using Viterbi
Using CRF
Examples
Technical Background
Literature
Related Topics

A conditional random field (CRF) is a type of discriminative, undirected probabilistic graphical model. A linear-chain CRF is a special type of CRF that assumes the current state depends only on the previous state.

Feature extraction modules are provided for text-analysis tasks such as part-of-speech (POS) tagging and named-entity resolution (NER). Currently, six feature types are implemented:

Edge Feature: transition feature that encodes the transition feature weight from current label to next label.
Start Feature: fired when the current token is the first token in a sequence.
End Feature: fired when the current token is the last token in a sequence.
Word Feature: fired when the current token is observed in the trained dictionary.
Unknown Feature: fired when the current token is not observed in the trained dictionary for at least a certain number of times (default 1).
Regex Feature: fired when the current token can be matched by a regular expression.

A Viterbi implementation is also provided to get the best label sequence and the conditional probability \( \Pr( \text{best label sequence} \mid \text{sequence}) \).

Following steps are required for CRF Learning and Inference:

Training Feature Generation
CRF Training
Testing Feature Generation
Inference using Viterbi

Training Feature Generation

The function takes train_segment_tbl and regex_tbl as input and does feature generation generating three tables dictionary_tbl, train_feature_tbl and train_featureset_tbl, that are required as an input for CRF training.

crf_train_fgen(train_segment_tbl,
               regex_tbl,
               label_tbl,
               dictionary_tbl,
               train_feature_tbl,
               train_featureset_tbl)

Arguments

train_segment_tbl

TEXT. Name of the training segment table. The table is expected to have the following columns:

doc_id	INTEGER. Document id column
start_pos	INTEGER. Index of a particular term in the respective document
seg_text	TEXT. Term at the respective `start_pos` in the document
label	INTEGER. Label id for the term corresponding to the actual label from `label_tbl`

regex_tbl

TEXT. Name of the regular expression table. The table is expected to have the following columns:

pattern	TEXT. Regular Expression
name	TEXT. Regular Expression name

label_tbl

TEXT. Name of the table containing unique labels and their id's. The table is expected to have the following columns:

id	INTEGER. Unique label id. NOTE: Must range from 0 to total number of labels in the table - 1.
label	TEXT. Label name

dictionary_tbl

TEXT. Name of the dictionary table to be created containing unique terms along with their counts. The table will have the following columns:

token	TEXT. Contains all the unique terms found in `train_segment_tbl`
total	INTEGER. Respective counts for the terms

train_feature_tbl

TEXT. Name of the training feature table to be created. The table will have the following columns:

doc_id	INTEGER. Document id
f_size	INTEGER. Feature set size. This value will be same for all the tuples in the table
sparse_r	DOUBLE PRECISION[]. Array union of individual single state features (previous label, label, feature index, start position, training existance indicator), ordered by their start position.
dense_m	DOUBLE PRECISION[]. Array union of (previous label, label, feature index, start position, training existance indicator) of edge features ordered by start position.
sparse_m	DOUBLE PRECISION[]. Array union of (feature index, previous label, label) of edge features ordered by feature index.

train_featureset_tbl

TEXT. Name of the table to be created containing distinct featuresets generated from training feature extraction. The table will have the following columns:

f_index	INTEGER. Column containing distinct featureset ids
f_name	TEXT. Feature name
feature	ARRAY. Feature value. The value is of the form [L1, L2] - If L1 = -1: represents single state feature with L2 being the current label id. - If L1 != -1: represents transition feature with L1 be the previous label and L2 be the current label.

Linear Chain CRF Training Function: The function takes train_feature_tbl and train_featureset_tbl tables generated in the training feature generation steps as input along with other required parameters and produces two output tables crf_stats_tbl and crf_weights_tbl.

lincrf_train(train_feature_tbl,
             train_featureset_tbl,
             label_tbl,
             crf_stats_tbl,
             crf_weights_tbl
             max_iterations
            )

Arguments

train_feature_tbl

TEXT. Name of the feature table generated during training feature generation

train_featureset_tbl

TEXT. Name of the featureset table generated during training feature generation

label_tbl

TEXT. Name of the label table used

crf_stats_table

TEXT. Name of the table to be created containing statistics for CRF training. The table has the following columns:

coef	DOUBLE PRECISION[]. Array of coefficients
log_likelihood	DOUBLE. Log-likelihood
num_iterations	INTEGER. The number of iterations at which the algorithm terminated

crf_weights_table

TEXT. Name of the table to be created creating learned feature weights. The table has the following columns:

id	INTEGER. Feature set id
name	TEXT. Feature name
prev_label_id	INTEGER. Label for the previous token encountered
label_id	INTEGER. Label of the token with the respective feature
weight	DOUBLE PRECISION. Weight for the respective feature set

max_iterations

INTEGER. The maximum number of iterations

Testing Feature Generation

crf_test_fgen(test_segment_tbl,
              dictionary_tbl,
              label_tbl,
              regex_tbl,
              crf_weights_tbl,
              viterbi_mtbl,
              viterbi_rtbl
             )

Arguments

test_segment_tbl

TEXT. Name of the testing segment table. The table is expected to have the following columns:

doc_id	INTEGER. Document id column
start_pos	INTEGER. Index of a particular term in the respective document
seg_text	TEXT. Term at the respective `start_pos` in the document

dictionary_tbl

TEXT. Name of the dictionary table created during training feature generation (crf_train_fgen)

label_tbl

TEXT. Name of the label table

regex_tbl

TEXT. Name of the regular expression table

crf_weights_tbl

TEXT. Name of the weights table generated during CRF training (lincrf_train)

viterbi_mtbl

TEXT. Name of the Viterbi M table to be created

viterbi_rtbl

TEXT. Name of the Viterbi R table to be created

Inference using Viterbi

vcrf_label(test_segment_tbl,
           viterbi_mtbl,
           viterbi_rtbl,
           label_tbl,
           result_tbl)

Arguments

test_segment_tbl: TEXT. Name of the testing segment table. For required table schema, please refer to arguments in previous section
viterbi_mtbl: TEXT. Name of the table viterbi_mtbl generated from testing feature generation crf_test_fgen.
viterbi_rtbl: TEXT. Name of the table viterbi_rtbl generated from testing feature generation crf_test_fgen.
label_tbl: TEXT. Name of the label table.
result_tbl: TEXT. Name of the result table to be created containing extracted best label sequences.

Using CRF

Generate text features, calculate their weights, and output the best label sequence for test data:

Perform feature generation on training data i.e. train_segment_tbl generating train_feature_tbl and train_featureset_tbl.

SELECT madlib.crf_train_fgen(
         'train_segment_tbl',
         'regex_tbl',
         'label_tbl',
         'dictionary_tbl',
         'train_feature_tbl',
         'train_featureset_tbl');

Use linear-chain CRF for training providing train_feature_tbl and train_featureset_tbl generated from previous step as an input.

SELECT madlib.lincrf_train(
         'train_feature_tbl',
         'train_featureset_tbl',
         'label_tbl',
         'crf_stats_tbl',
         'crf_weights_tbl',
         max_iterations);

Perform feature generation on testing data test_segment_tbl generating viterbi_mtbl and viterbi_rtbl required for inferencing.

SELECT madlib.crf_test_fgen(
         'test_segment_tbl',
         'dictionary_tbl',
         'label_tbl',
         'regex_tbl',
         'crf_weights_tbl',
         'viterbi_mtbl',
         'viterbi_rtbl');

Run the Viterbi function to get the best label sequence and the conditional probability \( \Pr( \text{best label sequence} \mid \text{sequence}) \).

SELECT madlib.vcrf_label(
         'test_segment_tbl',
         'viterbi_mtbl',
         'viterbi_rtbl',
         'label_tbl',
         'result_tbl');

Examples: This example uses a trivial training and test data set.

Load the label table, the regular expressions table, and the training segment table:

SELECT * FROM crf_label ORDER BY id;

Result:

 id | label
 ---+-------
  0 | #
  1 | $
  2 | ''
...
  8 | CC
  9 | CD
 10 | DT
 11 | EX
 12 | FW
 13 | IN
 14 | JJ
...

The regular expressions table:

SELECT * from crf_regex;

    pattern    |         name
 --------------+----------------------
 ^.+ing$       | endsWithIng
 ^[A-Z][a-z]+$ | InitCapital
 ^[A-Z]+$      | isAllCapital
 ^.*[0-9]+.*$  | containsDigit
...

The training segment table:

SELECT * from train_segmenttbl ORDER BY doc_id, start_pos;

 doc_id | start_pos |  seg_text  | label
 -------+-----------+------------+-------
      0 |         0 | Confidence |    18
      0 |         1 | in         |    13
      0 |         2 | the        |    10
      0 |         3 | pound      |    18
      0 |         4 | is         |    38
      0 |         5 | widely     |    26
...
      1 |         0 | Chancellor |    19
      1 |         1 | of         |    13
      1 |         2 | the        |    10
      1 |         3 | Exchequer  |    19
      1 |         4 | Nigel      |    19
...

Generate the training features:

SELECT crf_train_fgen( 'train_segmenttbl',
                       'crf_regex',
                       'crf_label',
                       'crf_dictionary',
                       'train_featuretbl',
                       'train_featureset'
                     );
SELECT * from crf_dictionary;

Result:

     token       | total
 ----------------+-------
 Hawthorne       |     1
 Mercedes-Benzes |     1
 Wolf            |     3
 best-known      |     1
 hairline        |     1
 accepting       |     2
 purchases       |    14
 trash           |     5
 co-venture      |     1
 restaurants     |     7
...

SELECT * from train_featuretbl;

Result:

 doc_id | f_size |            sparse_r           |             dense_m             |       sparse_m
 -------+--------+-------------------------------+---------------------------------+-----------------------
      2 |     87 | {-1,13,12,0,1,-1,13,9,0,1,..} | {13,31,79,1,1,31,29,70,2,1,...} | {51,26,2,69,29,17,...}
      1 |     87 | {-1,13,0,0,1,-1,13,9,0,1,...} | {13,0,62,1,1,0,13,54,2,1,13,..} | {51,26,2,69,29,17,...}

SELECT * from train_featureset;

 f_index |    f_name     | feature
 --------+---------------+---------
       1 | R_endsWithED  | {-1,29}
      13 | W_outweigh    | {-1,26}
      29 | U             | {-1,5}
      31 | U             | {-1,29}
      33 | U             | {-1,12}
      35 | W_a           | {-1,2}
      37 | W_possible    | {-1,6}
      15 | W_signaled    | {-1,29}
      17 | End.          | {-1,43}
      49 | W_'s          | {-1,16}
      63 | W_acquire     | {-1,26}
      51 | E.            | {26,2}
      69 | E.            | {29,17}
      71 | E.            | {2,11}
      83 | W_the         | {-1,2}
      85 | E.            | {16,11}
       4 | W_return      | {-1,11}
...

Train using linear CRF:

SELECT lincrf_train( 'train_featuretbl',
                     'train_featureset',
                     'crf_label',
                     'crf_stats_tbl',
                     'crf_weights_tbl',
                     20
             );

                                lincrf_train
 -----------------------------------------------------------------------------------
 CRF Train successful. Results stored in the specified CRF stats and weights table
 lincrf

View the feature weight table.

SELECT * from crf_weights_tbl;

Result:

 id |     name      | prev_label_id | label_id |      weight
 ---+---------------+---------------+----------+-------------------
  1 | R_endsWithED  |            -1 |       29 |  1.54128249293937
 13 | W_outweigh    |            -1 |       26 |  1.70691232223653
 29 | U             |            -1 |        5 |  1.40708515869008
 31 | U             |            -1 |       29 | 0.830356200936407
 33 | U             |            -1 |       12 | 0.769587378281239
 35 | W_a           |            -1 |        2 |  2.68470625883726
 37 | W_possible    |            -1 |        6 |  3.41773107604468
 15 | W_signaled    |            -1 |       29 |  1.68187039165771
 17 | End.          |            -1 |       43 |  3.07687845517082
 49 | W_'s          |            -1 |       16 |  2.61430312229883
 63 | W_acquire     |            -1 |       26 |  1.67247047385797
 51 | E.            |            26 |        2 |   3.0114240119435
 69 | E.            |            29 |       17 |  2.82385531733866
 71 | E.            |             2 |       11 |  3.00970493772732
 83 | W_the         |            -1 |        2 |  2.58742315259326
...

To find the best labels for a test set using the trained linear CRF model, repeat steps #1-2 and generate the test features, except instead of creating a new dictionary, use the dictionary generated from the training set.

SELECT * from test_segmenttbl ORDER BY doc_id, start_pos;

Result:

 doc_id | start_pos |   seg_text
 -------+-----------+---------------
      0 |         0 | Rockwell
      0 |         1 | International
      0 |         2 | Corp.
      0 |         3 | 's
      0 |         4 | Tulsa
      0 |         5 | unit
      0 |         6 | said
...
      1 |         0 | Rockwell
      1 |         1 | said
      1 |         2 | the
      1 |         3 | agreement
      1 |         4 | calls
...

SELECT crf_test_fgen( 'test_segmenttbl',
                      'crf_dictionary',
                      'crf_label',
                      'crf_regex',
                      'crf_weights_tbl',
                      'viterbi_mtbl',
                      'viterbi_rtbl'
                    );

Calculate the best label sequence and save in the table extracted_best_labels.

SELECT vcrf_label( 'test_segmenttbl',
                   'viterbi_mtbl',
                   'viterbi_rtbl',
                   'crf_label',
                   'extracted_best_labels'
                 );

View the best labels.

SELECT * FROM extracted_best_labels;

Result:

 doc_id | start_pos |   seg_text    | label | id | max_pos |   prob
 -------+-----------+---------------+-------+----+---------+----------
      0 |         0 | Rockwell      | NNP   | 19 |      27 | 0.000269
      0 |         1 | International | NNP   | 19 |      27 | 0.000269
      0 |         2 | Corp.         | NNP   | 19 |      27 | 0.000269
      0 |         3 | 's            | NNP   | 19 |      27 | 0.000269
...
      1 |         0 | Rockwell      | NNP   | 19 |      16 | 0.000168
      1 |         1 | said          | NNP   | 19 |      16 | 0.000168
      1 |         2 | the           | DT    | 10 |      16 | 0.000168
      1 |         3 | agreement     | JJ    | 14 |      16 | 0.000168
      1 |         4 | calls         | NNS   | 21 |      16 | 0.000168
...

Technical Background

Specifically, a linear-chain CRF is a distribution defined by

\[ p_\lambda(\boldsymbol y | \boldsymbol x) = \frac{\exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y)}}{Z_\lambda(\boldsymbol x)} \,. \]

where

\( F_m(\boldsymbol x, \boldsymbol y) = \sum_{i=1}^n f_m(y_i,y_{i-1},x_i) \) is a global feature function that is a sum along a sequence \( \boldsymbol x \) of length \( n \)
\( f_m(y_i,y_{i-1},x_i) \) is a local feature function dependent on the current token label \( y_i \), the previous token label \( y_{i-1} \), and the observation \( x_i \)
\( \lambda_m \) is the corresponding feature weight
\( Z_\lambda(\boldsymbol x) \) is an instance-specific normalizer
\[ Z_\lambda(\boldsymbol x) = \sum_{\boldsymbol y'} \exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y')} \]

A linear-chain CRF estimates the weights \( \lambda_m \) by maximizing the log-likelihood of a given training set \( T=\{(x_k,y_k)\}_{k=1}^N \).

The log-likelihood is defined as

\[ \ell_{\lambda}=\sum_k \log p_\lambda(y_k|x_k) =\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] \]

and the zero of its gradient

\[ \nabla \ell_{\lambda}=\sum_k[F(x_k,y_k)-E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] \]

is found since the maximum likelihood is reached when the empirical average of the global feature vector equals its model expectation. The MADlib implementation uses limited-memory BFGS (L-BFGS), a limited-memory variation of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) update, a quasi-Newton method for unconstrained optimization.

\(E_{p_\lambda(Y|x)}[F(x,Y)]\) is found by using a variant of the forward-backward algorithm:

\[ E_{p_\lambda(Y|x)}[F(x,Y)] = \sum_y p_\lambda(y|x)F(x,y) = \sum_i\frac{\alpha_{i-1}(f_i*M_i)\beta_i^T}{Z_\lambda(x)} \]

\[ Z_\lambda(x) = \alpha_n.1^T \]

where \(\alpha_i\) and \( \beta_i\) are the forward and backward state cost vectors defined by

\[ \alpha_i = \begin{cases} \alpha_{i-1}M_i, & 0<i<=n\\ 1, & i=0 \end{cases}\\ \]

\[ \beta_i^T = \begin{cases} M_{i+1}\beta_{i+1}^T, & 1<=i<n\\ 1, & i=n \end{cases} \]

To avoid overfitting, we penalize the likelihood with a spherical Gaussian weight prior:

\[ \ell_{\lambda}^\prime=\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] - \frac{\lVert \lambda \rVert^2}{2\sigma ^2} \]

\[ \nabla \ell_{\lambda}^\prime=\sum_k[F(x_k,y_k) - E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] - \frac{\lambda}{\sigma ^2} \]

Literature: [1] F. Sha, F. Pereira. Shallow Parsing with Conditional Random Fields, http://www-bcf.usc.edu/~feisha/pubs/shallow03.pdf

[2] Wikipedia, Conditional Random Field, http://en.wikipedia.org/wiki/Conditional_random_field

[3] A. Jaiswal, S.Tawari, I. Mansuri, K. Mittal, C. Tiwari (2012), CRF, http://crf.sourceforge.net/

[4] D. Wang, ViterbiCRF, http://www.cs.berkeley.edu/~daisyw/ViterbiCRF.html

[5] Wikipedia, Viterbi Algorithm, http://en.wikipedia.org/wiki/Viterbi_algorithm

[6] J. Nocedal. Updating Quasi-Newton Matrices with Limited Storage (1980), Mathematics of Computation 35, pp. 773-782

[7] J. Nocedal, Software for Large-scale Unconstrained Optimization, http://users.eecs.northwestern.edu/~nocedal/lbfgs.html

Related Topics

File crf.sql_in crf_feature_gen.sql_in viterbi.sql_in (documenting the SQL functions)