1.18.0 User Documentation for Apache MADlib
Support Vector Machines

Support vector machines are models for regression and classification tasks. SVM models have two particularly desirable features: robustness in the presence of noisy data and applicability to a variety of data configurations. At its core, a linear SVM model is a hyperplane separating two distinct classes of data (in the case of classification problems), in such a way that the distance between the hyperplane and the nearest training data point (called the margin) is maximized. Vectors that lie on this margin are called support vectors. With the support vectors fixed, perturbations of vectors beyond the margin will not affect the model; this contributes to the model’s robustness. By substituting a kernel function for the usual inner product, one can approximate a large variety of decision boundaries in addition to linear hyperplanes.

Classification Training Function
The SVM binary classification training function has the following format:
svm_classification(
source_table,
model_table,
dependent_varname,
independent_varname,
kernel_func,
kernel_params,
grouping_col,
params,
verbose
)

Arguments
source_table

TEXT. Name of the table containing the training data.

model_table

TEXT. Name of the output table containing the model. Details of the output tables are provided below.

dependent_varname

TEXT. Name of the dependent variable column. For classification, this column can contain values of any type, but must assume exactly two distinct values since only binary classification is currently supported.

independent_varname

TEXT. Expression list to evaluate for the independent variables. An intercept variable should not be included as part of this expression. See 'fit_intercept' in the kernel params for info on intercepts. Please note that expression should be able to be cast to DOUBLE PRECISION[].

kernel_func (optional)

TEXT, default: 'linear'. Type of kernel. Currently three kernel types are supported: 'linear', 'gaussian', and 'polynomial'. The text can be any subset of the three strings; for e.g., kernel_func='ga' will create a Gaussian kernel.

kernel_params (optional)

TEXT, defaults: NULL. Parameters for non-linear kernel in a comma-separated string of key-value pairs. The actual parameters differ depending on the value of kernel_func. See the description below for details.

grouping_col (optional)

TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, which results in running one model per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single model is generated. Please note that cross validation is not supported if grouping is used.

params (optional)

TEXT, default: NULL. Parameters for optimization and regularization in a comma-separated string of key-value pairs. If a list of values is provided, then cross-validation will be performed to select the best value from the list. See the description below for details.

verbose (optional)
BOOLEAN default: FALSE. Verbose output of the results of training.

Output tables
The model table produced by SVM contains the following columns:

coef FLOAT8. Vector of coefficients. TEXT Identifies the group to which the datum belongs. BIGINT. Numbers of rows processed. BIGINT. Numbers of rows skipped due to missing values or failures. INTEGER. Number of iterations completed by stochastic gradient descent algorithm. The algorithm either converged in this number of iterations or hit the maximum number specified in the optimization parameters. FLOAT8. Value of the objective function of SVM, expressed as an average loss per row over the source_table. See Technical Background section below for more details. FLOAT8. Value of the L2-norm of the (sub)-gradient of the objective function. TEXT[]. Vector of dependent variable labels. The first entry corresponds to -1 and the second to +1. For internal use only.

An auxiliary table named <model_table>_random is created if the kernel is not linear. It contains data needed to embed test data into a random feature space (see references [2,3]). This data is used internally by svm_predict and not meaningful on its own to the user, so you can ignore it.

A summary table named <model_table>_summary is also created, which has the following columns:

method 'svm' Version of MADlib which was used to generate the model. The data source table name. The model table name. The dependent variable. The independent variables. The kernel function. The kernel parameters, as well as random feature map data. Columns on which to group. A string containing the optimization parameters. A string containing the regularization parameters. Number of groups in SVM training. Number of failed groups in SVM training. Total numbers of rows processed in all groups. Total numbers of rows skipped in all groups due to missing values or failures.

If cross validation is used, a table is created with a user specified name having the following columns:

... Names of cross validation parameters Mean value of accuracy when predicted on the validation fold, averaged over all folds and all rows. Standard deviation of accuracy when predicted on the validation fold, averaged over all folds and all rows.

Regression Training Function
The SVM regression training function has the following format:
svm_regression(source_table,
model_table,
dependent_varname,
independent_varname,
kernel_func,
kernel_params,
grouping_col,
params,
verbose
)


Arguments

Specifications for regression are largely the same as for classification. In the model table, there is no dependent variable mapping. The following arguments have specifications which differ from svm_classification:

dependent_varname
TEXT. Name of the dependent variable column. For regression, this column can contain only values or expressions that can be cast to DOUBLE PRECISION. Otherwise, an error will be thrown.
params (optional)
TEXT, default: NULL. The parameters epsilon and eps_table are only meaningful for regression. See description below for more details.

Novelty Detection Training Function
The novelty detection function is a one-class SVM classifier, and has the following format:
svm_one_class(
source_table,
model_table,
independent_varname,
kernel_func,
kernel_params,
grouping_col,
params,
verbose
)

Arguments

Specifications for novelty detection are largely the same as for classification, except the dependent variable name is not specified. The model table is the same as that for classification.

Kernel Parameters
Kernel parameters are supplied in a string containing a comma-delimited list of name-value pairs. All of these named parameters are optional, and their order does not matter. You must use the format "<param_name> = <value>" to specify the value of a parameter, otherwise the parameter is ignored.
Parameters common to all kernels
fit_intercept
Default: True. The parameter fit_intercept is an indicator to add an intercept to the independent_varname array expression. The intercept is added to the end of the feature list - thus the last element of the coefficient list is the intercept.
n_components
Default: max(100, 2*num_features). The dimensionality of the transformed feature space. A larger value lowers the variance of the estimate of the kernel but requires more memory and takes longer to train.
Note
Setting the n_components kernel parameter properly is important to generate an accurate decision boundary and can make the difference between a good model and a useless model. Try increasing the value of n_components if you are not getting an accurate decision boundary. This parameter arises from using the primal formulation, in which we map data into a relatively low-dimensional randomized feature space [2, 3]. The parameter n_components is the dimension of that feature space. We use the primal in MADlib to support scaling to large data sets, compared to R or other single node implementations that use the dual formulation and hence do not have this type of mapping, since the the dimensionality of the transformed feature space in the dual is effectively infinite.
random_state
Default: 1. Seed used by a random number generator.
Parameters for 'gaussian' kernel
gamma
Default: 1/num_features. The parameter $$\gamma$$ in the Radius Basis Function kernel, i.e., $$\exp(-\gamma||x-y||^2)$$. Choosing a proper value for gamma is critical to the performance of kernel machine; e.g., while a large gamma tends to cause overfitting, a small gamma will make the model too constrained to capture the complexity of the data.
Parameters for 'polynomial' kernel
coef0
Default: 1.0. The independent term $$q$$ in $$(\langle x,y\rangle + q)^r$$. Must be larger than or equal to 0. When it is 0, the polynomial kernel is in homogeneous form.
degree
Default: 3. The parameter $$r$$ in $$(\langle x,y\rangle + q)^r$$.

Other Parameters
Parameters in this section are supplied in the params argument as a string containing a comma-delimited list of name-value pairs. All of these named parameters are optional, and their order does not matter. You must use the format "<param_name> = <value>" to specify the value of a parameter, otherwise the parameter is ignored.

Hyperparameter optimization can be carried out using the built-in cross validation mechanism, which is activated by assigning a value greater than 1 to the parameter n_folds in params. Please note that cross validation is not supported if grouping is used.

The values of a parameter to cross validate should be provided in a list. For example, if one wanted to regularize with the L1 norm and use a lambda value from the set {0.3, 0.4, 0.5}, one might input 'lambda={0.3, 0.4, 0.5}, norm=L1, n_folds=10' in params. Note that the use of '{}' and '[]' are both valid here.

Note
Note that not all of the parameters below can be cross-validated. For parameters where cross validation is allowed, their default values are presented in list format; e.g., [0.01].
  'init_stepsize = <value>,
decay_factor = <value>,
max_iter = <value>,
tolerance = <value>,
lambda = <value>,
norm = <value>,
epsilon = <value>,
eps_table = <value>,
validation_result = <value>,
n_folds = <value>,
class_weight = <value>'


Parameters

init_stepsize

Default: [0.01]. Also known as the initial learning rate. A small value is usually desirable to ensure convergence, while a large value provides more room for progress during training. Since the best value depends on the condition number of the data, in practice one often searches in an exponential grid using built-in cross validation; e.g., "init_stepsize = [1, 0.1, 0.001]". To reduce training time, it is common to run cross validation on a subsampled dataset, since this usually provides a good estimate of the condition number of the whole dataset. Then the resulting init_stepsize can be run on the whole dataset.

decay_factor

Default: [0.9]. Control the learning rate schedule: 0 means constant rate; <-1 means inverse scaling, i.e., stepsize = init_stepsize / iteration; > 0 means <exponential decay, i.e., stepsize = init_stepsize * decay_factor^iteration.

max_iter

Default: [100]. The maximum number of iterations allowed.

tolerance

Default: 1e-10. The criterion to end iterations. The training stops whenever the difference between the training models of two consecutive iterations is smaller than tolerance or the iteration number is larger than max_iter.

lambda

Default: [0.01]. Regularization parameter. Must be non-negative.

norm

Default: 'L2'. Name of the regularization, either 'L2' or 'L1'.

epsilon

Default: [0.01]. Determines the $$\epsilon$$ for $$\epsilon$$-regression. Ignored during classification. When training the model, differences of less than $$\epsilon$$ between estimated labels and actual labels are ignored. A larger $$\epsilon$$ will yield a model with fewer support vectors, but will not generalize as well to future data. Generally, it has been suggested that epsilon should increase with noisier data, and decrease with the number of samples. See [5].

eps_table

Default: NULL. Name of the input table that contains values of epsilon for different groups. Ignored when grouping_col is NULL. Define this input table if you want different epsilon values for different groups. The table consists of a column named epsilon which specifies the epsilon values, and one or more columns for grouping_col. Extra groups are ignored, and groups not present in this table will use the epsilon value specified in parameter epsilon.

validation_result

Default: NULL. Name of the table to store the cross validation scores. This table is only created if the name is not NULL. The cross validation scores are the mean and standard deviation of the accuracy when predicted on the validation fold, averaged over all folds and all rows. For classification, the accuracy metric used is the ratio of correct classifications. For regression, the accuracy metric used is the negative of mean squared error (negative to make it a concave problem, thus selecting max means the highest accuracy).

n_folds

Default: 0. Number of folds (k). Must be at least 2 to activate cross validation. If a value of k > 2 is specified, each fold is then used as a validation set once, while the other k - 1 folds form the training set.

class_weight

Default: NULL for classification, 'balanced' for one-class novelty detection, this param is not applicable for regression.

Set the weight for the classes. If not given (empty/NULL), all classes are set to have equal weight. If 'class_weight = balanced', values of y are automatically adjusted as inversely proportional to class frequencies in the input data i.e. the weights are set as n_samples / (2 * bincount(y)).

Alternatively, 'class_weight' can be a mapping, giving the weight for each class. E.g., for dependent variable values 'a' and 'b', the 'class_weight' might be {a: 1, b: 3}. This gives three times the weight to observations with class value 'b' compared to 'a'. (In the SVM algorithm, this translates into observations with class value 'b' contributing 3x to learning in the stochastic gradient step compared to 'a'.)

For regression, the class weights are always one.

Prediction Function
The prediction function is used to estimate the conditional mean given a new predictor. The same syntax is used for classification, regression and novelty detection:
svm_predict(model_table,
new_data_table,
id_col_name,
output_table)


Arguments

model_table

TEXT. Model table produced by the training function.

new_data_table

TEXT. Name of the table containing the prediction data. This table is expected to contain the same features that were used during training. The table should also contain id_col_name used for identifying each row.

id_col_name

TEXT. The name of the id column in the input table.

output_table
TEXT. Name of the table where output predictions are written. If this table name is already in use, then an error is returned. Table contains:
id Gives the 'id' for each prediction, corresponding to each row from the new_data_table. Provides the prediction for each row in new_data_table. For regression this would be the same as decision_function. For classification, this will be one of the dependent variable values. Provides the distance between each point and the separating hyperplane.

Examples

#### Classification

1. Create an input data set.
DROP TABLE IF EXISTS houses;
CREATE TABLE houses (id INT, tax INT, bedroom INT, bath FLOAT, price INT,
size INT, lot INT);
INSERT INTO houses VALUES
(1 ,  590 ,       2 ,    1 ,  50000 ,  770 , 22100),
(2 , 1050 ,       3 ,    2 ,  85000 , 1410 , 12000),
(3 ,   20 ,       3 ,    1 ,  22500 , 1060 ,  3500),
(4 ,  870 ,       2 ,    2 ,  90000 , 1300 , 17500),
(5 , 1320 ,       3 ,    2 , 133000 , 1500 , 30000),
(6 , 1350 ,       2 ,    1 ,  90500 ,  820 , 25700),
(7 , 2790 ,       3 ,  2.5 , 260000 , 2130 , 25000),
(8 ,  680 ,       2 ,    1 , 142500 , 1170 , 22000),
(9 , 1840 ,       3 ,    2 , 160000 , 1500 , 19000),
(10 , 3680 ,       4 ,    2 , 240000 , 2790 , 20000),
(11 , 1660 ,       3 ,    1 ,  87000 , 1030 , 17500),
(12 , 1620 ,       3 ,    2 , 118600 , 1250 , 20000),
(13 , 3100 ,       3 ,    2 , 140000 , 1760 , 38000),
(14 , 2070 ,       2 ,    3 , 148000 , 1550 , 14000),
(15 ,  650 ,       3 ,  1.5 ,  65000 , 1450 , 12000);

2. Train linear classification model and view the model. Categorical variable is price < $100,0000. DROP TABLE IF EXISTS houses_svm, houses_svm_summary; SELECT madlib.svm_classification('houses', 'houses_svm', 'price < 100000', 'ARRAY[1, tax, bath, size]' ); -- Set extended display on for easier reading of output \x on SELECT * FROM houses_svm;  -[ RECORD 1 ]------+-------------------------------------------------------------------------------- coef | {0.103994021495116,-0.00288252192097756,0.0540748706580464,0.00131729978010033} loss | 0.928463796644648 norm_of_gradient | 7849.34910604307 num_iterations | 100 num_rows_processed | 15 num_rows_skipped | 0 dep_var_mapping | {f,t}  3. Predict using linear model. We want to predict if house price is less than$100,000. We use the training data set for prediction as well, which is not usual but serves to show the syntax. The predicted results are in the prediction column and the actual data is in the actual column.
DROP TABLE IF EXISTS houses_pred;
'houses',
'id',
'houses_pred');
\x off
SELECT *, price < 100000 AS actual FROM houses JOIN houses_pred USING (id) ORDER BY id;

  id | tax  | bedroom | bath | price  | size |  lot  | prediction | decision_function  | actual
----+------+---------+------+--------+------+-------+------------+--------------------+--------
1 |  590 |       2 |    1 |  50000 |  770 | 22100 | t          |  0.211310440574799 | t
2 | 1050 |       3 |    2 |  85000 | 1410 | 12000 | t          |   0.37546191651855 | t
3 |   20 |       3 |    1 |  22500 | 1060 |  3500 | t          |    2.4021783278516 | t
4 |  870 |       2 |    2 |  90000 | 1300 | 17500 | t          |   0.63967342411632 | t
5 | 1320 |       3 |    2 | 133000 | 1500 | 30000 | f          | -0.179964783767855 | f
6 | 1350 |       2 |    1 |  90500 |  820 | 25700 | f          |  -1.78347623159173 | t
7 | 2790 |       3 |  2.5 | 260000 | 2130 | 25000 | f          |  -2.86795504439645 | f
8 |  680 |       2 |    1 | 142500 | 1170 | 22000 | t          |  0.811108105668757 | f
9 | 1840 |       3 |    2 | 160000 | 1500 | 19000 | f          |  -1.61739505790168 | f
10 | 3680 |       4 |    2 | 240000 | 2790 | 20000 | f          |  -3.96700444824078 | f
11 | 1660 |       3 |    1 |  87000 | 1030 | 17500 | f          |  -2.19489938920329 | t
12 | 1620 |       3 |    2 | 118600 | 1250 | 20000 | f          |  -1.53961627668269 | f
13 | 3100 |       3 |    2 | 140000 | 1760 | 38000 | f          |  -4.54881979553637 | f
14 | 2070 |       2 |    3 | 148000 | 1550 | 14000 | f          |  -2.06911803381861 | f
15 |  650 |       3 |  1.5 |  65000 | 1450 | 12000 | t          |   1.52704061329968 | t
(15 rows)

Count the miss-classifications:
SELECT COUNT(*) FROM houses_pred JOIN houses USING (id)
WHERE houses_pred.prediction != (houses.price < 100000);

 count
-------+
3

4. Train using Gaussian kernel. This time we specify the initial step size and maximum number of iterations to run. As part of the kernel parameter, we choose 10 as the dimension of the space where we train SVM. As a result, the model will be a 10 dimensional vector, instead of 4 as in the case of linear model.
DROP TABLE IF EXISTS houses_svm_gaussian, houses_svm_gaussian_summary, houses_svm_gaussian_random;
'houses_svm_gaussian',
'price < 100000',
'ARRAY[1, tax, bath, size]',
'gaussian',
'n_components=10',
'',
'init_stepsize=1, max_iter=200'
);
\x on
SELECT * FROM houses_svm_gaussian;

-[ RECORD 1 ]------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
coef               | {-1.67275666209207,1.5191640881642,-0.503066422926727,1.33250956564454,2.23009854231314,-0.0602475029497936,1.97466397155921,2.3668779833279,0.577739846910355,2.81255996089824}
loss               | 0.0571869097340991
num_iterations     | 177
num_rows_processed | 15
num_rows_skipped   | 0
dep_var_mapping    | {f,t}

5. Prediction using the Gaussian model. The predicted results are in the prediction column and the actual data is in the actual column.
DROP TABLE IF EXISTS houses_pred_gaussian;
'houses',
'id',
'houses_pred_gaussian');
\x off
SELECT *, price < 100000 AS actual FROM houses JOIN houses_pred_gaussian USING (id) ORDER BY id;

 id | tax  | bedroom | bath | price  | size |  lot  | prediction | decision_function  | actual
----+------+---------+------+--------+------+-------+------------+--------------------+--------
1 |  590 |       2 |    1 |  50000 |  770 | 22100 | t          |   1.89855833083557 | t
2 | 1050 |       3 |    2 |  85000 | 1410 | 12000 | t          |   1.47736856649617 | t
3 |   20 |       3 |    1 |  22500 | 1060 |  3500 | t          |  0.999999992995691 | t
4 |  870 |       2 |    2 |  90000 | 1300 | 17500 | t          |  0.999999989634351 | t
5 | 1320 |       3 |    2 | 133000 | 1500 | 30000 | f          |  -1.03645694166465 | f
6 | 1350 |       2 |    1 |  90500 |  820 | 25700 | t          |   1.16430515664766 | t
7 | 2790 |       3 |  2.5 | 260000 | 2130 | 25000 | f          | -0.545622670134529 | f
8 |  680 |       2 |    1 | 142500 | 1170 | 22000 | f          |  -1.00000000207512 | f
9 | 1840 |       3 |    2 | 160000 | 1500 | 19000 | f          |   -1.4748622470053 | f
10 | 3680 |       4 |    2 | 240000 | 2790 | 20000 | f          |  -1.00085274698056 | f
11 | 1660 |       3 |    1 |  87000 | 1030 | 17500 | t          |    1.8614251155696 | t
12 | 1620 |       3 |    2 | 118600 | 1250 | 20000 | f          |  -1.77616417509695 | f
13 | 3100 |       3 |    2 | 140000 | 1760 | 38000 | f          |  -1.07759348149549 | f
14 | 2070 |       2 |    3 | 148000 | 1550 | 14000 | f          |  -3.42475835116536 | f
15 |  650 |       3 |  1.5 |  65000 | 1450 | 12000 | t          |   1.00000008401961 | t
(15 rows)

Count the miss-classifications. Note this produces a more accurate result than the linear case for this data set:
SELECT COUNT(*) FROM houses_pred_gaussian JOIN houses USING (id)
WHERE houses_pred_gaussian.prediction != (houses.price < 100000);

 count
-------+
0
(1 row)

6. In the case of an unbalanced class-size dataset, use the 'balanced' parameter to classify when building the model:
DROP TABLE IF EXISTS houses_svm_gaussian, houses_svm_gaussian_summary, houses_svm_gaussian_random;
'houses_svm_gaussian',
'price < 150000',
'ARRAY[1, tax, bath, size]',
'gaussian',
'n_components=10',
'',
'init_stepsize=1, max_iter=200, class_weight=balanced'
);
\x on
SELECT * FROM houses_svm_gaussian;

-[ RECORD 1 ]------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
coef               | {0.891926151039837,0.169282494673541,-2.26539133689874,0.526518499596676,-0.900664505989526,0.508112011288015,-0.355474591147659,1.23127975981665,1.53694964239487,1.46496058633682}
loss               | 0.56900274445785
num_iterations     | 183
num_rows_processed | 15
num_rows_skipped   | 0
dep_var_mapping    | {f,t}


#### Regression

1. Create input data set. For regression we use part of the well known abalone data set https://archive.ics.uci.edu/ml/datasets/abalone :
DROP TABLE IF EXISTS abalone;
CREATE TABLE abalone (id INT, sex TEXT, length FLOAT, diameter FLOAT, height FLOAT, rings INT);
INSERT INTO abalone VALUES
(1,'M',0.455,0.365,0.095,15),
(2,'M',0.35,0.265,0.09,7),
(3,'F',0.53,0.42,0.135,9),
(4,'M',0.44,0.365,0.125,10),
(5,'I',0.33,0.255,0.08,7),
(6,'I',0.425,0.3,0.095,8),
(7,'F',0.53,0.415,0.15,20),
(8,'F',0.545,0.425,0.125,16),
(9,'M',0.475,0.37,0.125,9),
(10,'F',0.55,0.44,0.15,19),
(11,'F',0.525,0.38,0.14,14),
(12,'M',0.43,0.35,0.11,10),
(13,'M',0.49,0.38,0.135,11),
(14,'F',0.535,0.405,0.145,10),
(15,'F',0.47,0.355,0.1,10),
(16,'M',0.5,0.4,0.13,12),
(17,'I',0.355,0.28,0.085,7),
(18,'F',0.44,0.34,0.1,10),
(19,'M',0.365,0.295,0.08,7),
(20,'M',0.45,0.32,0.1,9);

2. Train a linear regression model:
DROP TABLE IF EXISTS abalone_svm_regression, abalone_svm_regression_summary;
'abalone_svm_regression',
'rings',
'ARRAY[1, length, diameter, height]'
);
\x on
SELECT * FROM abalone_svm_regression;

-[ RECORD 1 ]------+-----------------------------------------------------------------------
coef               | {1.998949892503,0.918517478913099,0.712125856084095,0.229379472956877}
loss               | 8.29033295818392
num_iterations     | 100
num_rows_processed | 20
num_rows_skipped   | 0
dep_var_mapping    | {NULL}

3. Predict using the linear regression model:
DROP TABLE IF EXISTS abalone_regr;
'abalone',
'id',
'abalone_regr');
\x off
SELECT * FROM abalone JOIN abalone_regr USING (id) ORDER BY id;

 id | sex | length | diameter | height | rings |    prediction    | decision_function
----+-----+--------+----------+--------+-------+------------------+-------------------
1 | M   |  0.455 |    0.365 |  0.095 |    15 | 2.69859240928376 |  2.69859240928376
2 | M   |   0.35 |    0.265 |   0.09 |     7 | 2.52978857282818 |  2.52978857282818
3 | F   |   0.53 |     0.42 |  0.135 |     9 | 2.81582333426116 |  2.81582333426116
4 | M   |   0.44 |    0.365 |  0.125 |    10 | 2.69169603073001 |  2.69169603073001
5 | I   |   0.33 |    0.255 |   0.08 |     7 | 2.50200316683054 |  2.50200316683054
6 | I   |  0.425 |      0.3 |  0.095 |     8 | 2.62474869654157 |  2.62474869654157
7 | F   |   0.53 |    0.415 |   0.15 |    20 | 2.81570339722408 |  2.81570339722408
8 | F   |  0.545 |    0.425 |  0.125 |    16 | 2.83086793257882 |  2.83086793257882
9 | M   |  0.475 |     0.37 |  0.125 |     9 | 2.72740477577673 |  2.72740477577673
10 | F   |   0.55 |     0.44 |   0.15 |    19 |  2.8518768970598 |   2.8518768970598
11 | F   |  0.525 |     0.38 |   0.14 |    14 | 2.78389260680315 |  2.78389260680315
12 | M   |   0.43 |     0.35 |   0.11 |    10 | 2.66838827339779 |  2.66838827339779
13 | M   |   0.49 |     0.38 |  0.135 |    11 | 2.75059759385832 |  2.75059759385832
14 | F   |  0.535 |    0.405 |  0.145 |    10 | 2.81202782833915 |  2.81202782833915
15 | F   |   0.47 |    0.355 |    0.1 |    10 | 2.70639581129576 |  2.70639581129576
16 | M   |    0.5 |      0.4 |   0.13 |    12 | 2.77287839069521 |  2.77287839069521
17 | I   |  0.355 |     0.28 |  0.085 |     7 | 2.54391615211472 |  2.54391615211472
18 | F   |   0.44 |     0.34 |    0.1 |    10 | 2.66815839489651 |  2.66815839489651
19 | M   |  0.365 |    0.295 |   0.08 |     7 | 2.56263631931732 |  2.56263631931732
20 | M   |   0.45 |     0.32 |    0.1 |     9 | 2.66310105219146 |  2.66310105219146
(20 rows)

RMS error:
SELECT SQRT(AVG((rings-prediction)*(rings-prediction))) as rms_error FROM abalone
JOIN abalone_regr USING (id);

    rms_error
-----------------+
9.0884271818321
(1 row)

4. Train a non-linear regression model using a Gaussian kernel:
DROP TABLE IF EXISTS abalone_svm_gaussian_regression, abalone_svm_gaussian_regression_summary, abalone_svm_gaussian_regression_random;
'abalone_svm_gaussian_regression',
'rings',
'ARRAY[1, length, diameter, height]',
'gaussian',
'n_components=10',
'',
'init_stepsize=1, max_iter=200'
);
\x on
SELECT * FROM abalone_svm_gaussian_regression;

-[ RECORD 1 ]------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
coef               | {4.49016341280977,2.19062972461334,-2.04673653356154,1.11216153651262,2.83478599238881,-4.23122821845785,4.17684533744501,-5.36892552740644,0.775782561685621,-3.62606941016707}
loss               | 2.66850539541894
num_iterations     | 163
num_rows_processed | 20
num_rows_skipped   | 0
dep_var_mapping    | {NULL}

5. Predict using Gaussian regression model:
DROP TABLE IF EXISTS abalone_gaussian_regr;
'abalone',
'id',
'abalone_gaussian_regr');
\x off
SELECT * FROM abalone JOIN abalone_gaussian_regr USING (id) ORDER BY id;

 id | sex | length | diameter | height | rings |    prediction    | decision_function
----+-----+--------+----------+--------+-------+------------------+-------------------
1 | M   |  0.455 |    0.365 |  0.095 |    15 | 9.92189555675422 |  9.92189555675422
2 | M   |   0.35 |    0.265 |   0.09 |     7 | 9.81553107620013 |  9.81553107620013
3 | F   |   0.53 |     0.42 |  0.135 |     9 | 10.0847384862759 |  10.0847384862759
4 | M   |   0.44 |    0.365 |  0.125 |    10 | 10.0100000075406 |  10.0100000075406
5 | I   |   0.33 |    0.255 |   0.08 |     7 | 9.74093262454458 |  9.74093262454458
6 | I   |  0.425 |      0.3 |  0.095 |     8 | 9.94807651709641 |  9.94807651709641
7 | F   |   0.53 |    0.415 |   0.15 |    20 | 10.1448936105369 |  10.1448936105369
8 | F   |  0.545 |    0.425 |  0.125 |    16 | 10.0579420659954 |  10.0579420659954
9 | M   |  0.475 |     0.37 |  0.125 |     9 |  10.055724626407 |   10.055724626407
10 | F   |   0.55 |     0.44 |   0.15 |    19 | 10.1225030222559 |  10.1225030222559
11 | F   |  0.525 |     0.38 |   0.14 |    14 |  10.160706707435 |   10.160706707435
12 | M   |   0.43 |     0.35 |   0.11 |    10 | 9.95760174386841 |  9.95760174386841
13 | M   |   0.49 |     0.38 |  0.135 |    11 | 10.0981242315617 |  10.0981242315617
14 | F   |  0.535 |    0.405 |  0.145 |    10 | 10.1501121415596 |  10.1501121415596
15 | F   |   0.47 |    0.355 |    0.1 |    10 | 9.97689437628973 |  9.97689437628973
16 | M   |    0.5 |      0.4 |   0.13 |    12 | 10.0633271219326 |  10.0633271219326
17 | I   |  0.355 |     0.28 |  0.085 |     7 | 9.79492924255328 |  9.79492924255328
18 | F   |   0.44 |     0.34 |    0.1 |    10 | 9.94856833428783 |  9.94856833428783
19 | M   |  0.365 |    0.295 |   0.08 |     7 | 9.78278863173308 |  9.78278863173308
20 | M   |   0.45 |     0.32 |    0.1 |     9 | 9.98822477687532 |  9.98822477687532
(20 rows)

Compute the RMS error. Note this produces a more accurate result than the linear case for this data set:
SELECT SQRT(AVG((rings-prediction)*(rings-prediction))) as rms_error FROM abalone
JOIN abalone_gaussian_regr USING (id);

    rms_error
------------------+
3.83678516581768
(1 row)

6. Cross validation. Let's run cross validation for different initial step sizes and lambda values:
DROP TABLE IF EXISTS abalone_svm_gaussian_regression, abalone_svm_gaussian_regression_summary,
abalone_svm_gaussian_regression_random, abalone_svm_gaussian_regression_cv;
'abalone_svm_gaussian_regression',
'rings',
'ARRAY[1, length, diameter, height]',
'gaussian',
'n_components=10',
'',
'init_stepsize=[0.01,1], n_folds=3, max_iter=200, lambda=[0.01, 0.1, 0.5],
validation_result=abalone_svm_gaussian_regression_cv'
);
\x on
SELECT * FROM abalone_svm_gaussian_regression;

-[ RECORD 1 ]------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
coef               | {4.46074154389204,2.19335800415975,-2.14775901092668,1.06805891149535,2.91168496475457,-3.95521278459095,4.20496790233169,-5.28144330907061,0.427743633754918,-3.58999505728692}
loss               | 2.68317592175908
num_iterations     | 169
num_rows_processed | 20
num_rows_skipped   | 0
dep_var_mapping    | {NULL}

View the summary table showing the final model parameters are those that produced the lowest error in the cross validation runs:
SELECT * FROM abalone_svm_gaussian_regression_summary;

-[ RECORD 1 ]--------+------------------------------------------------------------------------------------
method               | SVR
version_number       | 1.18.0
source_table         | abalone
model_table          | abalone_svm_gaussian_regression
dependent_varname    | rings
independent_varname  | ARRAY[1, length, diameter, height]
kernel_func          | gaussian
kernel_params        | gamma=0.25, n_components=10,random_state=1, fit_intercept=False, fit_in_memory=True
grouping_col         | NULL
optim_params         | init_stepsize=1.0,
| decay_factor=0.9,
| max_iter=200,
| tolerance=1e-10,
| epsilon=0.01,
| eps_table=,
| class_weight=
reg_params           | lambda=0.01, norm=l2, n_folds=3
num_all_groups       | 1
num_failed_groups    | 0
total_rows_processed | 20
total_rows_skipped   | 0
(6 rows)

View the statistics for the various cross validation values:
\x off
SELECT * FROM abalone_svm_gaussian_regression_cv;

 init_stepsize | lambda |   mean_score   | std_dev_score
---------------+--------+----------------+----------------
1.0 |   0.01 | -4.06711568585 | 0.435966381366
1.0 |    0.1 | -4.08068428345 |  0.44660797513
1.0 |    0.5 | -4.52576046087 |  0.20597876382
0.01 |   0.01 | -11.0231044189 | 0.739956548721
0.01 |    0.1 | -11.0244799274 | 0.740029346709
0.01 |    0.5 | -11.0305445077 | 0.740350338532
(6 rows)

7. Predict using the cross-validated Gaussian regression model:
DROP TABLE IF EXISTS abalone_gaussian_regr;
'abalone',
'id',
'abalone_gaussian_regr');

Compute the RMS error. Note this produces a more accurate result than the previous run with the Gaussian kernel:
SELECT SQRT(AVG((rings-prediction)*(rings-prediction))) as rms_error FROM abalone
JOIN abalone_gaussian_regr USING (id);

    rms_error
------------------+
3.84208909699442
(1 row)


#### Novelty Detection

1. Now train a non-linear one-class SVM for novelty detection, using a Gaussian kernel. Note that the dependent variable is not a parameter for one-class:
DROP TABLE IF EXISTS houses_one_class_gaussian, houses_one_class_gaussian_summary, houses_one_class_gaussian_random;
'houses_one_class_gaussian',
'ARRAY[1,tax,bedroom,bath,size,lot,price]',
'gaussian',
'gamma=0.5,n_components=55, random_state=3',
NULL,
'max_iter=100, init_stepsize=10,lambda=10, tolerance=0'
);
\x on
SELECT * FROM houses_one_class_gaussian;

Result:
-[ RECORD 1 ]------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
coef               | {redacted for brevity}
loss               | 0.944016313708205
num_iterations     | 100
num_rows_processed | 16
num_rows_skipped   | -1
dep_var_mapping    | {-1,1}

2. For the novelty detection using one-class, let's create a test data set using the last 3 values from the training set plus an outlier at the end (10x price):
DROP TABLE IF EXISTS houses_one_class_test;
CREATE TABLE houses_one_class_test (id INT, tax INT, bedroom INT, bath FLOAT, price INT,
size INT, lot INT);
INSERT INTO houses_one_class_test VALUES
(1 , 3100 ,       3 ,    2 , 140000 , 1760 , 38000),
(2 , 2070 ,       2 ,    3 , 148000 , 1550 , 14000),
(3 ,  650 ,       3 ,  1.5 ,  65000 , 1450 , 12000),
(4 ,  650 ,       3 ,  1.5 ,  650000 , 1450 , 12000);

Now run prediction on the Gaussian one-class novelty detection model. Result shows the last row predicted to be novel:
DROP TABLE IF EXISTS houses_pred;
'houses_one_class_test',
'id',
'houses_pred');
\x off
SELECT * FROM houses_one_class_test JOIN houses_pred USING (id) ORDER BY id;

Result showing the last row predicted to be novel:
 id | tax  | bedroom | bath | price  | size |  lot  | prediction |  decision_function
----+------+---------+------+--------+------+-------+------------+---------------------
1 | 3100 |       3 |    2 | 140000 | 1760 | 38000 |          1 |   0.111497008121437
2 | 2070 |       2 |    3 | 148000 | 1550 | 14000 |          1 |  0.0996021345169148
3 |  650 |       3 |  1.5 |  65000 | 1450 | 12000 |          1 |  0.0435064008756942
4 |  650 |       3 |  1.5 | 650000 | 1450 | 12000 |         -1 | -0.0168967845338403


Technical Background

To solve linear SVM, the following objective function is minimized:

$\underset{w,b}{\text{Minimize }} \lambda||w||^2 + \frac{1}{n}\sum_{i=1}^n \ell(y_i,f_{w,b}(x_i))$

where $$(x_1,y_1),\ldots,(x_n,y_n)$$ are labeled training data and $$\ell(y,f(x))$$ is a loss function. When performing classification, $$\ell(y,f(x)) = \max(0,1-yf(x))$$ is the hinge loss. For regression, the loss function $$\ell(y,f(x)) = \max(0,|y-f(x)|-\epsilon)$$ is used.

If $$f_{w,b}(x) = \langle w, x\rangle + b$$ is linear, then the objective function is convex and incremental gradient descent (IGD, or SGD) can be applied to find a global minimum. See Feng, et al. [1] for more details.

To learn with Gaussian or polynomial kernels, the training data is first mapped via a random feature map in such a way that the usual inner product in the feature space approximates the kernel function in the input space. The linear SVM training function is then run on the resulting data. See the papers [2,3] for more information on random feature maps.

Also, see the book [4] by Scholkopf and Smola for more details on SVMs in general.

Literature

[1] Xixuan Feng, Arun Kumar, Ben Recht, and Christopher Re: Towards a Unified Architecture for in-RDBMS analytics, in SIGMOD Conference, 2012 http://www.eecs.berkeley.edu/~brecht/papers/12.FengEtAl.SIGMOD.pdf

[2] Purushottam Kar and Harish Karnick: Random Feature Maps for Dot Product Kernels, Proceedings of the 15th International Conference on Artificial Intelligence and Statistics, 2012, http://machinelearning.wustl.edu/mlpapers/paper_files/AISTATS2012_KarK12.pdf

[3] Ali Rahmini and Ben Recht: Random Features for Large-Scale Kernel Machines, Neural Information Processing Systems 2007, http://www.eecs.berkeley.edu/~brecht/papers/07.rah.rec.nips.pdf

[4] Bernhard Scholkopf and Alexander Smola: Learning with Kernels, The MIT Press, Cambridge, MA, 2002.

[5] Vladimir Cherkassky and Yunqian Ma: Practical Selection of SVM Parameters and Noise Estimation for SVM Regression, Neural Networks, 2004 http://www.ece.umn.edu/users/cherkass/N2002-SI-SVM-13-whole.pdf

Related Topics

File svm.sql_in documenting the training function