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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><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 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 binary 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 since only binary classification is currently supported. </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, expressed as an average loss per row over the <em>source_table</em>. 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>If cross validation is used, a table is created with a user specified name having the following columns: </p><table class="output">
<tr>
<th>... </th><td>Names of cross validation parameters </td></tr>
<tr>
<th>mean_score </th><td>Mean value of accuracy when predicted on the validation fold, averaged over all folds and all rows. </td></tr>
<tr>
<th>std_dev_score </th><td>Standard deviation of accuracy when predicted on the validation fold, averaged over all folds and all rows. </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: 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. <dl class="section note"><dt>Note</dt><dd>Setting the <em>n_components</em> 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 <em>n_components</em> 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 <em>n_components</em> 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.</dd></dl>
</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 \(\gamma\) in the Radius Basis Function kernel, i.e., \(\exp(-\gamma||x-y||^2)\). 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 \(q\) 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. </dd>
<dt>degree </dt>
<dd>Default: 3. The parameter \(r\) in \( (\langle x,y\rangle + q)^r \). </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 the difference between the training models of two consecutive iterations is 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 \(\epsilon\) 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]. </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 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 <em>max</em> means the highest accuracy). </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: NULL for classification, 'balanced' for one-class novelty detection, this param is not applicable for regression.</p>
<p>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)).</p>
<p>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'.)</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></dd></dl>
<h4>Classification</h4>
<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);
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);
</pre></li>
<li>Train linear classification model and view the model. Categorical variable is price &lt; $100,0000. <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]'
);
-- Set extended display on for easier reading of output
\x on
SELECT * FROM houses_svm;
</pre> <pre class="result">
-[ 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}
</pre></li>
<li>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 <em>prediction</em> column and the actual data is in the <em>actual</em> column. <pre class="example">
DROP TABLE IF EXISTS houses_pred;
SELECT madlib.svm_predict('houses_svm',
'houses',
'id',
'houses_pred');
\x off
SELECT *, price &lt; 100000 AS actual FROM houses JOIN houses_pred USING (id) ORDER BY id;
</pre> <pre class="result">
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)
</pre> Count the miss-classifications: <pre class="example">
SELECT COUNT(*) FROM houses_pred JOIN houses USING (id)
WHERE houses_pred.prediction != (houses.price &lt; 100000);
</pre> <pre class="result">
count
-------+
3
</pre></li>
<li>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. <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'
);
\x on
SELECT * FROM houses_svm_gaussian;
</pre> <pre class="result">
-[ 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
norm_of_gradient | 1.18281830047046
num_iterations | 177
num_rows_processed | 15
num_rows_skipped | 0
dep_var_mapping | {f,t}
</pre></li>
<li>Prediction using the Gaussian model. The predicted results are in the <em>prediction</em> column and the actual data is in the <em>actual</em> column. <pre class="example">
DROP TABLE IF EXISTS houses_pred_gaussian;
SELECT madlib.svm_predict('houses_svm_gaussian',
'houses',
'id',
'houses_pred_gaussian');
\x off
SELECT *, price &lt; 100000 AS actual FROM houses JOIN houses_pred_gaussian USING (id) ORDER BY id;
</pre> <pre class="result">
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)
</pre> Count the miss-classifications. Note this produces a more accurate result than the linear case for this data set: <pre class="example">
SELECT COUNT(*) FROM houses_pred_gaussian JOIN houses USING (id)
WHERE houses_pred_gaussian.prediction != (houses.price &lt; 100000);
</pre> <pre class="result">
count
-------+
0
(1 row)
</pre></li>
<li>In the case of an unbalanced class-size dataset, use the 'balanced' parameter to classify when building the model: <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'
);
\x on
SELECT * FROM houses_svm_gaussian;
</pre> <pre class="result">
-[ 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
norm_of_gradient | 0.989597662458527
num_iterations | 183
num_rows_processed | 15
num_rows_skipped | 0
dep_var_mapping | {f,t}
</pre></li>
</ol>
<h4>Regression</h4>
<ol type="1">
<li>Create input data set. For regression we use part of the well known abalone data set <a href="https://archive.ics.uci.edu/ml/datasets/abalone">https://archive.ics.uci.edu/ml/datasets/abalone</a> : <pre class="example">
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);
</pre></li>
<li>Train a linear regression model: <pre class="example">
DROP TABLE IF EXISTS abalone_svm_regression, abalone_svm_regression_summary;
SELECT madlib.svm_regression('abalone',
'abalone_svm_regression',
'rings',
'ARRAY[1, length, diameter, height]'
);
\x on
SELECT * FROM abalone_svm_regression;
</pre> <pre class="result">
-[ RECORD 1 ]------+-----------------------------------------------------------------------
coef | {1.998949892503,0.918517478913099,0.712125856084095,0.229379472956877}
loss | 8.29033295818392
norm_of_gradient | 23.225177785827
num_iterations | 100
num_rows_processed | 20
num_rows_skipped | 0
dep_var_mapping | {NULL}
</pre></li>
<li>Predict using the linear regression model: <pre class="example">
DROP TABLE IF EXISTS abalone_regr;
SELECT madlib.svm_predict('abalone_svm_regression',
'abalone',
'id',
'abalone_regr');
\x off
SELECT * FROM abalone JOIN abalone_regr USING (id) ORDER BY id;
</pre> <pre class="result">
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)
</pre> RMS error: <pre class="example">
SELECT SQRT(AVG((rings-prediction)*(rings-prediction))) as rms_error FROM abalone
JOIN abalone_regr USING (id);
</pre> <pre class="result">
rms_error
-----------------+
9.0884271818321
(1 row)
</pre></li>
<li>Train a non-linear regression model using a Gaussian kernel: <pre class="example">DROP TABLE IF EXISTS abalone_svm_gaussian_regression, abalone_svm_gaussian_regression_summary, abalone_svm_gaussian_regression_random;
SELECT madlib.svm_regression( 'abalone',
'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;
</pre> <pre class="result">
-[ 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
norm_of_gradient | 0.97440079536379
num_iterations | 163
num_rows_processed | 20
num_rows_skipped | 0
dep_var_mapping | {NULL}
</pre></li>
<li>Predict using Gaussian regression model: <pre class="example">
DROP TABLE IF EXISTS abalone_gaussian_regr;
SELECT madlib.svm_predict('abalone_svm_gaussian_regression',
'abalone',
'id',
'abalone_gaussian_regr');
\x off
SELECT * FROM abalone JOIN abalone_gaussian_regr USING (id) ORDER BY id;
</pre> <pre class="result">
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)
</pre> Compute the RMS error. Note this produces a more accurate result than the linear case for this data set: <pre class="example">
SELECT SQRT(AVG((rings-prediction)*(rings-prediction))) as rms_error FROM abalone
JOIN abalone_gaussian_regr USING (id);
</pre> <pre class="result">
rms_error
------------------+
3.83678516581768
(1 row)
</pre></li>
<li>Cross validation. Let's run cross validation for different initial step sizes and lambda values: <pre class="example">
DROP TABLE IF EXISTS abalone_svm_gaussian_regression, abalone_svm_gaussian_regression_summary,
abalone_svm_gaussian_regression_random, abalone_svm_gaussian_regression_cv;
SELECT madlib.svm_regression( 'abalone',
'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;
</pre> <pre class="result">
-[ 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
norm_of_gradient | 0.69852112502746
num_iterations | 169
num_rows_processed | 20
num_rows_skipped | 0
dep_var_mapping | {NULL}
</pre> View the summary table showing the final model parameters are those that produced the lowest error in the cross validation runs: <pre class="example">
SELECT * FROM abalone_svm_gaussian_regression_summary;
</pre> <pre class="result">
-[ 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)
</pre> View the statistics for the various cross validation values: <pre class="example">
\x off
SELECT * FROM abalone_svm_gaussian_regression_cv;
</pre> <pre class="result">
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)
</pre></li>
<li>Predict using the cross-validated Gaussian regression model: <pre class="example">
DROP TABLE IF EXISTS abalone_gaussian_regr;
SELECT madlib.svm_predict('abalone_svm_gaussian_regression',
'abalone',
'id',
'abalone_gaussian_regr');
</pre> Compute the RMS error. Note this produces a more accurate result than the previous run with the Gaussian kernel: <pre class="example">
SELECT SQRT(AVG((rings-prediction)*(rings-prediction))) as rms_error FROM abalone
JOIN abalone_gaussian_regr USING (id);
</pre> <pre class="result">
rms_error
------------------+
3.84208909699442
(1 row)
</pre></li>
</ol>
<h4>Novelty Detection</h4>
<ol type="1">
<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'
);
\x on
SELECT * FROM houses_one_class_gaussian;
</pre> Result: <pre class="result">
-[ RECORD 1 ]------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
coef | {redacted for brevity}
loss | 0.944016313708205
norm_of_gradient | 14.5271059047443
num_iterations | 100
num_rows_processed | 16
num_rows_skipped | -1
dep_var_mapping | {-1,1}
</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);
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);
</pre> Now run prediction on the Gaussian one-class novelty detection model. Result shows the last row predicted to be novel: <pre class="example">
DROP TABLE IF EXISTS houses_pred;
SELECT madlib.svm_predict('houses_one_class_gaussian',
'houses_one_class_test',
'id',
'houses_pred');
\x off
SELECT * FROM houses_one_class_test JOIN houses_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>
</ol>
<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">
\[ \underset{w,b}{\text{Minimize }} \lambda||w||^2 + \frac{1}{n}\sum_{i=1}^n \ell(y_i,f_{w,b}(x_i)) \]
</p>
<p>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 <em>hinge loss</em>. For regression, the loss function \(\ell(y,f(x)) = \max(0,|y-f(x)|-\epsilon)\) is used.</p>
<p>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.</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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