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<title>MADlib: Support Vector Machines</title>
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<div class="title">Support Vector Machines<div class="ingroups"><a class="el" href="group__grp__super.html">Supervised Learning</a></div></div> </div>
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<div class="contents">
<div class="toc"><b>Contents</b></p><ul>
<li class="level1">
<a href="#svm_classification">Classification Function</a> </li>
<li class="level1">
<a href="#svm_regression">Regression Function</a> </li>
<li class="level1">
<a href="#novelty_detection">Novelty Detection</a> </li>
<li class="level1">
<a href="#kernel_params">Kernel Parameters</a> </li>
<li class="level1">
<a href="#parameters">Other Parameters</a> </li>
<li class="level1">
<a href="#predict">Prediction Functions</a> </li>
<li class="level1">
<a href="#example">Examples</a> </li>
<li class="level1">
<a href="#background">Technical Background</a> </li>
<li class="level1">
<a href="#literature">Literature</a> </li>
<li class="level1">
<a href="#related">Related Topics</a> </li>
</ul>
</div><p>Support Vector Machines (SVMs) 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 <em>linear</em> 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 <em>margin</em>) 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. <a class="anchor" id="svm_classification"></a></p><dl class="section user"><dt>Classification Training Function</dt><dd>The SVM classification training function has the following format: <pre class="syntax">
svm_classification(
source_table,
model_table,
dependent_varname,
independent_varname,
kernel_func,
kernel_params,
grouping_col,
params,
verbose
)
</pre> <b>Arguments</b> <dl class="arglist">
<dt>source_table </dt>
<dd><p class="startdd">TEXT. Name of the table containing the training data.</p>
<p class="enddd"></p>
</dd>
<dt>model_table </dt>
<dd><p class="startdd">TEXT. Name of the output table containing the model. Details of the output tables are provided below. </p>
<p class="enddd"></p>
</dd>
<dt>dependent_varname </dt>
<dd><p class="startdd">TEXT. Name of the dependent variable column. For classification, this column can contain values of any type, but must assume exactly two distinct values. Otherwise, an error will be thrown. </p>
<p class="enddd"></p>
</dd>
<dt>independent_varname </dt>
<dd><p class="startdd">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[].</p>
<p class="enddd"></p>
</dd>
<dt>kernel_func (optional) </dt>
<dd><p class="startdd">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. </p>
<p class="enddd"></p>
</dd>
<dt>kernel_params (optional) </dt>
<dd><p class="startdd">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 <em>kernel_func</em>. See the description below for details. </p>
<p class="enddd"></p>
</dd>
<dt>grouping_col (optional) </dt>
<dd><p class="startdd">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.</p>
<p class="enddd"></p>
</dd>
<dt>params (optional) </dt>
<dd><p class="startdd">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 <em>best</em> value from the list. See the description below for details. </p>
<p class="enddd"></p>
</dd>
<dt>verbose (optional) </dt>
<dd>BOOLEAN default: FALSE. Verbose output of the results of training. </dd>
</dl>
</dd></dl>
<p><b>Output tables</b> <br />
The model table produced by SVM contains the following columns: </p><table class="output">
<tr>
<th>coef </th><td>FLOAT8. Vector of coefficients. </td></tr>
<tr>
<th>grouping_key </th><td>TEXT Identifies the group to which the datum belongs. </td></tr>
<tr>
<th>num_rows_processed </th><td>BIGINT. Numbers of rows processed. </td></tr>
<tr>
<th>num_rows_skipped </th><td>BIGINT. Numbers of rows skipped due to missing values or failures. </td></tr>
<tr>
<th>num_iterations </th><td>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. </td></tr>
<tr>
<th>loss </th><td>FLOAT8. Value of the objective function of SVM. See Technical Background section below for more details. </td></tr>
<tr>
<th>norm_of_gradient </th><td>FLOAT8. Value of the L2-norm of the (sub)-gradient of the objective function. </td></tr>
<tr>
<th>__dep_var_mapping </th><td>TEXT[]. Vector of dependent variable labels. The first entry corresponds to -1 and the second to +1. For internal use only. </td></tr>
</table>
<p>An auxiliary table named &lt;model_table&gt;_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.</p>
<p>A summary table named &lt;model_table&gt;_summary is also created, which has the following columns: </p><table class="output">
<tr>
<th>method </th><td>'svm' </td></tr>
<tr>
<th>version_number </th><td>Version of MADlib which was used to generate the model. </td></tr>
<tr>
<th>source_table </th><td>The data source table name. </td></tr>
<tr>
<th>model_table </th><td>The model table name. </td></tr>
<tr>
<th>dependent_varname </th><td>The dependent variable. </td></tr>
<tr>
<th>independent_varname </th><td>The independent variables. </td></tr>
<tr>
<th>kernel_func </th><td>The kernel function. </td></tr>
<tr>
<th>kernel_parameters </th><td>The kernel parameters, as well as random feature map data. </td></tr>
<tr>
<th>grouping_col </th><td>Columns on which to group. </td></tr>
<tr>
<th>optim_params </th><td>A string containing the optimization parameters. </td></tr>
<tr>
<th>reg_params </th><td>A string containing the regularization parameters. </td></tr>
<tr>
<th>num_all_groups </th><td>Number of groups in SVM training. </td></tr>
<tr>
<th>num_failed_groups </th><td>Number of failed groups in SVM training. </td></tr>
<tr>
<th>total_rows_processed </th><td>Total numbers of rows processed in all groups. </td></tr>
<tr>
<th>total_rows_skipped </th><td>Total numbers of rows skipped in all groups due to missing values or failures. </td></tr>
</table>
<p><a class="anchor" id="svm_regression"></a></p><dl class="section user"><dt>Regression Training Function</dt><dd>The SVM regression training function has the following format: <pre class="syntax">
svm_regression(source_table,
model_table,
dependent_varname,
independent_varname,
kernel_func,
kernel_params,
grouping_col,
params,
verbose
)
</pre></dd></dl>
<p><b>Arguments</b> </p>
<p>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: </p><dl class="arglist">
<dt>dependent_varname </dt>
<dd>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. </dd>
<dt>params (optional) </dt>
<dd>TEXT, default: NULL. The parameters <em>epsilon</em> and <em>eps_table</em> are only meaningful for regression. See description below for more details. </dd>
</dl>
<p><a class="anchor" id="novelty_detection"></a></p><dl class="section user"><dt>Novelty Detection Training Function</dt><dd>The novelty detection function is a one-class SVM classifier, and has the following format: <pre class="syntax">
svm_one_class(
source_table,
model_table,
independent_varname,
kernel_func,
kernel_params,
grouping_col,
params,
verbose
)
</pre> <b>Arguments</b> </dd></dl>
<p>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.</p>
<p><a class="anchor" id="kernel_params"></a></p><dl class="section user"><dt>Kernel Parameters</dt><dd>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 "&lt;param_name&gt; = &lt;value&gt;" to specify the value of a parameter, otherwise the parameter is ignored.</dd></dl>
<dl class="arglist">
<dt><em>Parameters common to all kernels</em></dt>
<dd></dd>
<dt>fit_intercept </dt>
<dd>Default: True. The parameter <em>fit_intercept</em> is an indicator to add an intercept to the <em>independent_varname</em> array expression. The intercept is added to the end of the feature list - thus the last element of the coefficient list is the intercept. </dd>
<dt>n_components </dt>
<dd>Default: 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. </dd>
<dt>random_state </dt>
<dd>Default: 1. Seed used by a random number generator. </dd>
</dl>
<dl class="arglist">
<dt><em>Parameters for 'gaussian' kernel</em></dt>
<dd></dd>
<dt>gamma </dt>
<dd>Default: 1/num_features. The parameter <img class="formulaInl" alt="$\gamma$" src="form_517.png"/> in the Radius Basis Function kernel, i.e., <img class="formulaInl" alt="$\exp(-\gamma||x-y||^2)$" src="form_518.png"/>. Choosing a proper value for <em>gamma</em> is critical to the performance of kernel machine; e.g., while a large <em>gamma</em> tends to cause overfitting, a small <em>gamma</em> will make the model too constrained to capture the complexity of the data. </dd>
</dl>
<dl class="arglist">
<dt><em>Parameters for 'polynomial' kernel</em></dt>
<dd></dd>
<dt>coef0 </dt>
<dd>Default: 1.0. The independent term <img class="formulaInl" alt="$q$" src="form_519.png"/> in <img class="formulaInl" alt="$ (\langle x,y\rangle + q)^r $" src="form_520.png"/>. Must be larger than or equal to 0. When it is 0, the polynomial kernel is in homogeneous form. </dd>
<dt>degree </dt>
<dd>Default: 3. The parameter <img class="formulaInl" alt="$r$" src="form_521.png"/> in <img class="formulaInl" alt="$ (\langle x,y\rangle + q)^r $" src="form_520.png"/>. </dd>
</dl>
<p><a class="anchor" id="parameters"></a></p><dl class="section user"><dt>Other Parameters</dt><dd>Parameters in this section are supplied in the <em>params</em> 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 "&lt;param_name&gt; = &lt;value&gt;" to specify the value of a parameter, otherwise the parameter is ignored.</dd></dl>
<p>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 <em>n_folds</em> in <em>params</em>. Please note that cross validation is not supported if grouping is used.</p>
<p>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 <em>params</em>. Note that the use of '{}' and '[]' are both valid here. </p><dl class="section note"><dt>Note</dt><dd>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].</dd></dl>
<pre class="syntax">
'init_stepsize = &lt;value&gt;,
decay_factor = &lt;value&gt;,
max_iter = &lt;value&gt;,
tolerance = &lt;value&gt;,
lambda = &lt;value&gt;,
norm = &lt;value&gt;,
epsilon = &lt;value&gt;,
eps_table = &lt;value&gt;,
validation_result = &lt;value&gt;,
n_folds = &lt;value&gt;,
class_weight = &lt;value&gt;'
</pre><p> <b>Parameters</b> </p><dl class="arglist">
<dt>init_stepsize </dt>
<dd><p class="startdd">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 <em>init_stepsize</em> can be run on the whole dataset.</p>
<p></p>
<p class="enddd"></p>
</dd>
<dt>decay_factor </dt>
<dd><p class="startdd">Default: [0.9]. Control the learning rate schedule: 0 means constant rate; &lt;-1 means inverse scaling, i.e., stepsize = init_stepsize / iteration; &gt; 0 means &lt;exponential decay, i.e., stepsize = init_stepsize * decay_factor^iteration. </p>
<p class="enddd"></p>
</dd>
<dt>max_iter </dt>
<dd><p class="startdd">Default: [100]. The maximum number of iterations allowed. </p>
<p class="enddd"></p>
</dd>
<dt>tolerance </dt>
<dd><p class="startdd">Default: 1e-10. The criterion to end iterations. The training stops whenever &lt;the difference between the training models of two consecutive iterations is &lt;smaller than <em>tolerance</em> or the iteration number is larger than <em>max_iter</em>. </p>
<p class="enddd"></p>
</dd>
<dt>lambda </dt>
<dd><p class="startdd">Default: [0.01]. Regularization parameter. Must be non-negative. </p>
<p class="enddd"></p>
</dd>
<dt>norm </dt>
<dd><p class="startdd">Default: 'L2'. Name of the regularization, either 'L2' or 'L1'. </p>
<p class="enddd"></p>
</dd>
<dt>epsilon </dt>
<dd><p class="startdd">Default: [0.01]. Determines the <img class="formulaInl" alt="$\epsilon$" src="form_522.png"/> for <img class="formulaInl" alt="$\epsilon$" src="form_522.png"/>-regression. Ignored during classification. When training the model, differences of less than <img class="formulaInl" alt="$\epsilon$" src="form_522.png"/> between estimated labels and actual labels are ignored. A larger <img class="formulaInl" alt="$\epsilon$" src="form_522.png"/> 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]. </p>
<p class="enddd"></p>
</dd>
<dt>eps_table </dt>
<dd><p class="startdd">Default: NULL. Name of the input table that contains values of epsilon for different groups. Ignored when <em>grouping_col</em> is NULL. Define this input table if you want different epsilon values for different groups. The table consists of a column named <em>epsilon</em> which specifies the epsilon values, and one or more columns for <em>grouping_col</em>. Extra groups are ignored, and groups not present in this table will use the epsilon value specified in parameter <em>epsilon</em>. </p>
<p class="enddd"></p>
</dd>
<dt>validation_result </dt>
<dd><p class="startdd">Default: NULL. Name of the table to store the cross validation results including the values of parameters and their averaged error values. For now, simple metric like 0-1 loss is used for classification and mean square error is used for regression. The table is only created if the name is not NULL. </p>
<p class="enddd"></p>
</dd>
<dt>n_folds </dt>
<dd><p class="startdd">Default: 0. Number of folds (k). Must be at least 2 to activate cross validation. If a value of k &gt; 2 is specified, each fold is then used as a validation set once, while the other k - 1 folds form the training set. </p>
<p class="enddd"></p>
</dd>
<dt>class_weight </dt>
<dd><p class="startdd">Default: 1 for classification, 'balanced' for one-class novelty detection, n/a for regression.</p>
<p>Set the weight for the positive and negative classes. If not given, all classes are set to have weight one. 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 / (n_classes * bincount(y)).</p>
<p>Alternatively, class_weight can be a mapping, giving the weight for each class. Eg. For dependent variable values 'a' and 'b', the class_weight can be {a: 2, b: 3}. This would lead to each 'a' tuple's y value multiplied by 2 and each 'b' y value will be multiplied by 3.</p>
<p class="enddd">For regression, the class weights are always one. </p>
</dd>
</dl>
<p><a class="anchor" id="predict"></a></p><dl class="section user"><dt>Prediction Function</dt><dd>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: <pre class="syntax">
svm_predict(model_table,
new_data_table,
id_col_name,
output_table)
</pre></dd></dl>
<p><b>Arguments</b> </p><dl class="arglist">
<dt>model_table </dt>
<dd><p class="startdd">TEXT. Model table produced by the training function.</p>
<p class="enddd"></p>
</dd>
<dt>new_data_table </dt>
<dd><p class="startdd">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.</p>
<p class="enddd"></p>
</dd>
<dt>id_col_name </dt>
<dd><p class="startdd">TEXT. The name of the id column in the input table.</p>
<p class="enddd"></p>
</dd>
<dt>output_table </dt>
<dd>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: <table class="output">
<tr>
<th>id </th><td>Gives the 'id' for each prediction, corresponding to each row from the new_data_table. </td></tr>
<tr>
<th>prediction </th><td>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. </td></tr>
<tr>
<th>decision_function </th><td>Provides the distance between each point and the separating hyperplane. </td></tr>
</table>
</dd>
</dl>
<p><a class="anchor" id="example"></a></p><dl class="section user"><dt>Examples</dt><dd><ol type="1">
<li>Create an input data set. <pre class="example">
DROP TABLE IF EXISTS houses;
CREATE TABLE houses (id INT, tax INT, bedroom INT, bath FLOAT, price INT,
size INT, lot INT);
COPY houses FROM STDIN WITH DELIMITER '|';
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
\.
</pre></li>
<li>Train a classification model. First, use a linear model. <pre class="example">
DROP TABLE IF EXISTS houses_svm, houses_svm_summary;
SELECT madlib.svm_classification('houses',
'houses_svm',
'price &lt; 100000',
'ARRAY[1, tax, bath, size]'
);
</pre></li>
<li>View the result for the linear classification model. <pre class="example">
-- Set extended display on for easier reading of output
\x ON
SELECT * FROM houses_svm;
</pre> Result: <pre class="result">
-[ RECORD 1 ]------+---------------------------------------------------------------
coef | {0.152192069515,-0.29631947495,0.0968619000065,0.362682248051}
loss | 601.279740124
norm_of_gradient | 1300.96615851627
num_iterations | 100
num_rows_processed | 15
num_rows_skipped | 0
dep_var_mapping | {f,t}
</pre></li>
<li>Next generate a nonlinear model using a 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. A larger number will lead to a more powerful model but run the risk of overfitting. As a result, the model will be a 10 dimensional vector, instead of 4 as in the case of linear model, which we will verify when we examine the models. <pre class="example">
DROP TABLE IF EXISTS houses_svm_gaussian, houses_svm_gaussian_summary, houses_svm_gaussian_random;
SELECT madlib.svm_classification( 'houses',
'houses_svm_gaussian',
'price &lt; 100000',
'ARRAY[1, tax, bath, size]',
'gaussian',
'n_components=10',
'',
'init_stepsize=1, max_iter=200'
);
</pre></li>
<li>View the results from kernel SVM for classification. <pre class="example">
-- Set extended display on for easier reading of output
\x ON
SELECT * FROM houses_svm_gaussian;
</pre> Result: <pre class="result">
-[ RECORD 1 ]------+--------------------------------------------------------------------------------------------------------------------------------------------------
coef | {0.183800813574,-0.78724997813,1.54121854068,1.24432527042,4.01230959334,1.07061097224,-4.92576349408,0.437699542875,0.3128600981,-1.63880635658}
loss | 0.998735180388
norm_of_gradient | 0.729823950583579
num_iterations | 196
num_rows_processed | 15
num_rows_skipped | 0
dep_var_mapping | {f,t}
</pre></li>
<li>The regression models have a similar format (model output not shown). First, for a linear model: <pre class="example">
DROP TABLE IF EXISTS houses_svm_regression, houses_svm_regression_summary;
SELECT madlib.svm_regression('houses',
'houses_svm_regression',
'price',
'ARRAY[1, tax, bath, size]'
);
</pre> For a non-linear regression model using a Gaussian kernel: <pre class="example">
DROP TABLE IF EXISTS houses_svm_gaussian_regression, houses_svm_gaussian_regression_summary, houses_svm_gaussian_regression_random;
SELECT madlib.svm_regression( 'houses',
'houses_svm_gaussian_regression',
'price',
'ARRAY[1, tax, bath, size]',
'gaussian',
'n_components=10',
'',
'init_stepsize=1, max_iter=200'
);
</pre></li>
<li>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: <pre class="example">
DROP TABLE IF EXISTS houses_one_class_gaussian, houses_one_class_gaussian_summary, houses_one_class_gaussian_random;
select madlib.svm_one_class('houses',
'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'
);
</pre></li>
<li>View the result for the Gaussian novelty detection model. <pre class="example">
-- Set extended display on for easier reading of output
\x ON
SELECT * FROM houses_one_class_gaussian;
</pre> Result: <pre class="result">
-[ RECORD 1 ]------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
coef | {redacted for brevity}
loss | 15.1053343738
norm_of_gradient | 13.9133653663837
num_iterations | 100
num_rows_processed | 16
num_rows_skipped | -1
dep_var_mapping | {-1,1}
</pre></li>
<li>Now let's look at the prediction functions. We want to predict if house price is less than $100,000. In the following examples we will 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 <em>prediction</em> column and the actual data is in the <em>target</em> column. For the linear model: <pre class="example">
DROP TABLE IF EXISTS houses_pred;
SELECT madlib.svm_predict('houses_svm', 'houses', 'id', 'houses_pred');
SELECT *, price &lt; 100000 AS target FROM houses JOIN houses_pred USING (id) ORDER BY id;
</pre> Result: <pre class="result">
id | tax | bedroom | bath | price | size | lot | prediction | decision_function | target
----+------+---------+------+--------+------+-------+------------+--------------------+--------
1 | 590 | 2 | 1 | 50000 | 770 | 22100 | t | 104.685894748292 | t
2 | 1050 | 3 | 2 | 85000 | 1410 | 12000 | t | 200.592436923938 | t
3 | 20 | 3 | 1 | 22500 | 1060 | 3500 | t | 378.765847404582 | t
4 | 870 | 2 | 2 | 90000 | 1300 | 17500 | t | 214.034895129328 | t
5 | 1320 | 3 | 2 | 133000 | 1500 | 30000 | t | 153.227581012028 | f
6 | 1350 | 2 | 1 | 90500 | 820 | 25700 | f | -102.382793811158 | t
7 | 2790 | 3 | 2.5 | 260000 | 2130 | 25000 | f | -53.8237999423388 | f
8 | 680 | 2 | 1 | 142500 | 1170 | 22000 | t | 223.090041223192 | f
9 | 1840 | 3 | 2 | 160000 | 1500 | 19000 | f | -0.858545961972027 | f
10 | 3680 | 4 | 2 | 240000 | 2790 | 20000 | f | -78.226279884182 | f
11 | 1660 | 3 | 1 | 87000 | 1030 | 17500 | f | -118.078558954948 | t
12 | 1620 | 3 | 2 | 118600 | 1250 | 20000 | f | -26.3388234857219 | f
13 | 3100 | 3 | 2 | 140000 | 1760 | 38000 | f | -279.923699905712 | f
14 | 2070 | 2 | 3 | 148000 | 1550 | 14000 | f | -50.7810508979155 | f
15 | 650 | 3 | 1.5 | 65000 | 1450 | 12000 | t | 333.579085875975 | t
</pre> Prediction using the Gaussian model: <pre class="example">
DROP TABLE IF EXISTS houses_pred_gaussian;
SELECT madlib.svm_predict('houses_svm_gaussian', 'houses', 'id', 'houses_pred_gaussian');
SELECT *, price &lt; 100000 AS target FROM houses JOIN houses_pred_gaussian USING (id) ORDER BY id;
</pre> This produces a more accurate result than the linear case for this small data set: <pre class="result">
id | tax | bedroom | bath | price | size | lot | prediction | decision_function | target
----+------+---------+------+--------+------+-------+------------+-------------------+--------
1 | 590 | 2 | 1 | 50000 | 770 | 22100 | t | 1.00338548176312 | t
2 | 1050 | 3 | 2 | 85000 | 1410 | 12000 | t | 1.00000000098154 | t
3 | 20 | 3 | 1 | 22500 | 1060 | 3500 | t | 0.246566699635389 | t
4 | 870 | 2 | 2 | 90000 | 1300 | 17500 | t | 1.0000000003367 | t
5 | 1320 | 3 | 2 | 133000 | 1500 | 30000 | f | -1.98940593324397 | f
6 | 1350 | 2 | 1 | 90500 | 820 | 25700 | t | 3.74336995109761 | t
7 | 2790 | 3 | 2.5 | 260000 | 2130 | 25000 | f | -1.01574407296086 | f
8 | 680 | 2 | 1 | 142500 | 1170 | 22000 | f | -1.0000000002071 | f
9 | 1840 | 3 | 2 | 160000 | 1500 | 19000 | f | -3.88267069310101 | f
10 | 3680 | 4 | 2 | 240000 | 2790 | 20000 | f | -3.44507576539002 | f
11 | 1660 | 3 | 1 | 87000 | 1030 | 17500 | t | 2.3409866081761 | t
12 | 1620 | 3 | 2 | 118600 | 1250 | 20000 | f | -3.51563221173085 | f
13 | 3100 | 3 | 2 | 140000 | 1760 | 38000 | f | -1.00000000011163 | f
14 | 2070 | 2 | 3 | 148000 | 1550 | 14000 | f | -1.87710363254055 | f
15 | 650 | 3 | 1.5 | 65000 | 1450 | 12000 | t | 1.34334834982263 | t
</pre></li>
<li>Prediction using the linear regression model: <pre class="example">
DROP TABLE IF EXISTS houses_regr;
SELECT madlib.svm_predict('houses_svm_regression', 'houses', 'id', 'houses_regr');
SELECT * FROM houses JOIN houses_regr USING (id) ORDER BY id;
</pre> Result for the linear regression model: <pre class="result">
id | tax | bedroom | bath | price | size | lot | prediction | decision_function
----+------+---------+------+--------+------+-------+------------------+-------------------
1 | 590 | 2 | 1 | 50000 | 770 | 22100 | 55288.6992755623 | 55288.6992755623
2 | 1050 | 3 | 2 | 85000 | 1410 | 12000 | 99978.8137019119 | 99978.8137019119
3 | 20 | 3 | 1 | 22500 | 1060 | 3500 | 43157.5130381023 | 43157.5130381023
4 | 870 | 2 | 2 | 90000 | 1300 | 17500 | 88098.9557296729 | 88098.9557296729
5 | 1320 | 3 | 2 | 133000 | 1500 | 30000 | 114803.884262468 | 114803.884262468
6 | 1350 | 2 | 1 | 90500 | 820 | 25700 | 88899.5186193813 | 88899.5186193813
7 | 2790 | 3 | 2.5 | 260000 | 2130 | 25000 | 201108.397013076 | 201108.397013076
8 | 680 | 2 | 1 | 142500 | 1170 | 22000 | 75004.3236915733 | 75004.3236915733
9 | 1840 | 3 | 2 | 160000 | 1500 | 19000 | 136434.749667136 | 136434.749667136
10 | 3680 | 4 | 2 | 240000 | 2790 | 20000 | 264483.856987395 | 264483.856987395
11 | 1660 | 3 | 1 | 87000 | 1030 | 17500 | 110180.048139857 | 110180.048139857
12 | 1620 | 3 | 2 | 118600 | 1250 | 20000 | 117300.841695563 | 117300.841695563
13 | 3100 | 3 | 2 | 140000 | 1760 | 38000 | 199229.683967752 | 199229.683967752
14 | 2070 | 2 | 3 | 148000 | 1550 | 14000 | 147998.930271016 | 147998.930271016
15 | 650 | 3 | 1.5 | 65000 | 1450 | 12000 | 84936.7661235861 | 84936.7661235861
</pre> For the non-linear Gaussian regression model (output not shown): <pre class="example">
DROP TABLE IF EXISTS houses_gaussian_regr;
SELECT madlib.svm_predict('houses_svm_gaussian_regression', 'houses', 'id', 'houses_gaussian_regr');
SELECT * FROM houses JOIN houses_gaussian_regr USING (id) ORDER BY id;
</pre></li>
<li>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): <pre class="example">
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);
COPY houses_one_class_test FROM STDIN WITH DELIMITER '|';
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
\.
</pre> Now run prediction on the Gaussian one-class novelty detection model: <pre class="example">
DROP TABLE IF EXISTS houses_once_class_pred;
SELECT madlib.svm_predict('houses_one_class_gaussian', 'houses_one_class_test', 'id', 'houses_one_class_pred');
SELECT * FROM houses_one_class_test JOIN houses_one_class_pred USING (id) ORDER BY id;
</pre> Result showing the last row predicted to be novel: <pre class="result">
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
</pre></li>
<li>Create a model for an unbalanced class-size dataset, then use the 'balanced' parameter to classify: <pre class="example">
DROP TABLE IF EXISTS houses_svm_gaussian, houses_svm_gaussian_summary, houses_svm_gaussian_random;
SELECT madlib.svm_classification( 'houses',
'houses_svm_gaussian',
'price &lt; 150000',
'ARRAY[1, tax, bath, size]',
'gaussian',
'n_components=10',
'',
'init_stepsize=1, max_iter=200, class_weight=balanced'
);
SELECT * FROM houses_svm_gaussian;
</pre> <pre class="result">
-[ RECORD 1 ]------+----------------------------------------------------------------------------------------------------------------------------------------------------
coef | {-0.621843913637,2.4166374426,-1.54726833725,-1.74512599505,1.16231799548,-0.54019307285,-4.14373293694,-0.623069170717,3.59669949057,-1.005501237}
loss | 1.87657250199
norm_of_gradient | 1.41148000266816
num_iterations | 174
num_rows_processed | 15
num_rows_skipped | 0
dep_var_mapping | {f,t}
</pre> Note that the results you get for all examples may vary with the platform you are using.</li>
</ol>
</dd></dl>
<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
<p>To solve linear SVM, the following objective function is minimized: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ \underset{w,b}{\text{Minimize }} \lambda||w||^2 + \frac{1}{n}\sum_{i=1}^n \ell(y_i,f_{w,b}(x_i)) \]" src="form_523.png"/>
</p>
<p>where <img class="formulaInl" alt="$(x_1,y_1),\ldots,(x_n,y_n)$" src="form_524.png"/> are labeled training data and <img class="formulaInl" alt="$\ell(y,f(x))$" src="form_525.png"/> is a loss function. When performing classification, <img class="formulaInl" alt="$\ell(y,f(x)) = \max(0,1-yf(x))$" src="form_526.png"/> is the <em>hinge loss</em>. For regression, the loss function <img class="formulaInl" alt="$\ell(y,f(x)) = \max(0,|y-f(x)|-\epsilon)$" src="form_527.png"/> is used.</p>
<p>If <img class="formulaInl" alt="$ f_{w,b}(x) = \langle w, x\rangle + b$" src="form_528.png"/> 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.</p>
<p>To learn with Gaussian or polynomial kernels, the training data is first mapped via a <em>random feature map</em> 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.</p>
<p>Also, see the book [4] by Scholkopf and Smola for more details on SVMs in general.</p>
<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
<p><a class="anchor" id="svm-lit-1"></a>[1] Xixuan Feng, Arun Kumar, Ben Recht, and Christopher Re: Towards a Unified Architecture for in-RDBMS analytics, in SIGMOD Conference, 2012 <a href="http://www.eecs.berkeley.edu/~brecht/papers/12.FengEtAl.SIGMOD.pdf">http://www.eecs.berkeley.edu/~brecht/papers/12.FengEtAl.SIGMOD.pdf</a></p>
<p><a class="anchor" id="svm-lit-2"></a>[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, <a href="http://machinelearning.wustl.edu/mlpapers/paper_files/AISTATS2012_KarK12.pdf">http://machinelearning.wustl.edu/mlpapers/paper_files/AISTATS2012_KarK12.pdf</a></p>
<p><a class="anchor" id="svm-lit-3"></a>[3] Ali Rahmini and Ben Recht: Random Features for Large-Scale Kernel Machines, Neural Information Processing Systems 2007, <a href="http://www.eecs.berkeley.edu/~brecht/papers/07.rah.rec.nips.pdf">http://www.eecs.berkeley.edu/~brecht/papers/07.rah.rec.nips.pdf</a></p>
<p><a class="anchor" id="svm-lit-4"></a>[4] Bernhard Scholkopf and Alexander Smola: Learning with Kernels, The MIT Press, Cambridge, MA, 2002.</p>
<p><a class="anchor" id="svm-lit-5"></a>[5] Vladimir Cherkassky and Yunqian Ma: Practical Selection of SVM Parameters and Noise Estimation for SVM Regression, Neural Networks, 2004 <a href="http://www.ece.umn.edu/users/cherkass/N2002-SI-SVM-13-whole.pdf">http://www.ece.umn.edu/users/cherkass/N2002-SI-SVM-13-whole.pdf</a></p>
<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
<p>File <a class="el" href="svm_8sql__in.html" title="SQL functions for SVM (Poisson) ">svm.sql_in</a> documenting the training function</p>
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