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 * to you under the Apache License, Version 2.0 (the
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 * Unless required by applicable law or agreed to in writing,
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 * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
 * KIND, either express or implied.  See the License for the
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 *
 *
 * @file mlp.sql_in
 *
 * @brief SQL functions for multilayer perceptron
 * @date June 2012
 *
 *
 *//* ----------------------------------------------------------------------- */

m4_include(`SQLCommon.m4')

/**
@addtogroup grp_nn

<div class="toc"><b>Contents</b><ul>
<li class="level1"><a href="#mlp_classification">Classification</a></li>
<li class="level1"><a href="#mlp_regression">Regression</a></li>
<li class="level1"><a href="#optimizer_params">Optimizer 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>

Multilayer Perceptron (MLP) is a type of neural network that can be
used for regression and classification.

MLPs consist of several
fully connected hidden layers with non-linear activation
functions.  In the case of classification, the final layer of the
neural net has as many nodes as classes, and the output of the
neural net can be interpreted as the probability that a given input
feature belongs to a specific class.

MLP can be used with or without mini-batching.
The advantage of using mini-batching is that it
can perform better than stochastic gradient descent
(default MADlib optimizer)
because it uses more than one training example at a time,
typically resulting faster and smoother convergence [3].

@note
In order to use mini-batching, you must first run
the <a href="group__grp__minibatch__preprocessing.html">Mini-Batch Preprocessor</a>,
which is a utility that prepares input data for
use by models that support mini-batch as an optimization option,
such as MLP.  This is a one-time operation and you would only
need to re-run the preprocessor if your input data has changed,
or if you change the grouping parameter.

@brief Solves classification and regression problems with several
fully connected layers and non-linear activation functions.

@anchor mlp_classification
@par Classification Training Function
The MLP classification training function has the following format:

<pre class="syntax">
mlp_classification(
    source_table,
    output_table,
    independent_varname,
    dependent_varname,
    hidden_layer_sizes,
    optimizer_params,
    activation,
    weights,
    warm_start,
    verbose,
    grouping_col
    )
</pre>

\b Arguments
<dl class="arglist">
  <dt>source_table</dt>
  <dd>TEXT. Name of the table containing the training data.
  If you are using mini-batching, this is the name of the output
  table from the mini-batch preprocessor.</dd>

  <dt>output_table</dt>
  <dd>TEXT. Name of the output table containing the model. Details of the output
   table are shown below.
  </dd>

  <dt>independent_varname</dt>
  <dd>TEXT. Expression list to evaluate for the independent variables. It should
  be a numeric array expression.
  If you are using mini-batching, set this parameter to 'independent_varname'
  which is the hardcoded name of the column from the mini-batch preprocessor
  containing the packed independent variables.

  @note
  If you are not using mini-batching,
  please note that an intercept variable should not be included as part
  of this expression - this is different from other MADlib modules.  Also
  please note that independent variables should be encoded properly.
  All values are cast to DOUBLE PRECISION, so categorical variables should be
  one-hot or dummy encoded as appropriate.
  See <a href="group__grp__encode__categorical.html">Encoding Categorical Variables</a>
  for more details.
  </dd>

  <dt>dependent_varname</dt>
  <dd> TEXT. Name of the dependent variable column. For classification, supported types are:
  text, varchar, character varying, char, character
  integer, smallint, bigint, and boolean.
  If you are using mini-batching, set this parameter to 'dependent_varname'
  which is the hardcoded name of the column from the mini-batch preprocessor
  containing the packed dependent variables.</dd>

  <DT>hidden_layer_sizes (optional)</DT>
  <DD>INTEGER[], default: ARRAY[100].
  The number of neurons in each hidden layer.  The length of this array will
  determine the number of hidden layers.  For example, ARRAY[5,10] means 2 hidden
  layers, one with 5 neurons and the other with 10 neurons.
  Use ARRAY[]::INTEGER[] for no hidden layers.
  </DD>

  <DT>optimizer_params (optional)</DT>
  <DD>TEXT, default: NULL.
    Parameters for optimization in a comma-separated string
    of key-value pairs. See the description below for details.
  </DD>

  <DT>activation (optional)</DT>
  <DD>TEXT, default: 'sigmoid'.
    Activation function. Currently three functions are supported: 'sigmoid' (default),
    'relu', and 'tanh'. The text can be any prefix of the three
    strings; for e.g., specifying 's' will use sigmoid activation.
  </DD>

  <DT>weights (optional)</DT>
  <DD>TEXT, default: 1.
    Column name for giving different weights to different rows during training.
    E.g., a weight of two for a specific row is equivalent to dupicating that row.
    This weight is incorporated into the update during stochastic gradient
    descent (SGD), but is not be used for loss calculations. If not specified,
    weight for each row will default to 1 (equal weights).  Column should be a
    numeric type.
    @note
    The 'weights' parameter cannot be used if you use mini-batching of the source dataset.
  </DD>

  <DT>warm_start (optional)</DT>
  <DD>BOOLEAN, default: FALSE.
    Initalize neural network weights with the coefficients from the last call of the training
    function. If set to true, neural network weights will be initialized from the output_table
    generated by the previous run. Note that all parameters other than
    optimizer_params and verbose must remain constant between calls when
    warm_start is used.

    @note
    The warm start feature works based on the name of the output_table.
    When using warm start, do not drop the output table or the output table summary
    before calling the training function, since these are needed to obtain the
    neural network weights from the previous run.
    If you are not using warm start, the output table and the output table
    summary must be dropped in the usual way before calling the training function.

  </DD>

  <DT>verbose (optional)</DT>
  <DD>BOOLEAN, default: FALSE. Provides verbose output of the results of training,
  including the value of loss at each iteration.</DD>
  @note
    There are some subtleties on the reported per-iteration loss
    values because we are working in a distributed system.
    When mini-batching is used (i.e., batch gradient descent),
    loss per iteration is an average of losses across all mini-batches
    and epochs on a segment.  Losses across all segments then get
    averaged to give overall loss for the model for the iteration.
    This will tend to be a pessimistic estimate of loss.
    When mini-batching is not used (i.e., stochastic gradient descent),
    we use the model state from the previous iteration to compute the loss
    at the start of the current iteration on the whole data set.  This
    is an accurate computation of loss for the iteration.

  <DT>grouping_col (optional)</DT>
  <DD>TEXT, default: NULL.
    A single column or a list of comma-separated columns that divides the input
    data into discrete groups, resulting in one model per group. When this value
    is NULL, no grouping is used and a single model is generated for all data.
    If you are using mini-batching, you must have run the mini-batch
    preprocessor with exactly the same groups that you specify here for MLP
    training.  If you change the groups, or remove the groups, then you must re-
    run the mini-batch preprocessor.</dd>
  </DD>
</dl>

<b>Output tables</b>
<br>
    The model table produced by MLP contains the following columns:
    <table class="output">
      <tr>
        <th>coeffs</th>
        <td>FLOAT8[]. Flat array containing the weights of the neural net.</td>
      </tr>
      <tr>
        <th>n_iterations</th>
        <td>INTEGER. Number of iterations completed by the 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. The cross entropy loss over the training data.
        See Technical Background section below for more details.</td>
      </tr>
      <tr>
        <th>grouping columns</th>
        <td>If grouping_col is specified during training, a column for each grouping column
        is created.</td>
      </tr>
    </table>

A summary table named \<output_table\>_summary is also created, which has the following columns:
    <table class="output">
    <tr>
        <th>source_table</th>
        <td>The source table.</td>
    </tr>
    <tr>
        <th>independent_varname</th>
        <td>The independent variables.</td>
    </tr>
    <tr>
        <th>dependent_varname</th>
        <td>The dependent variable.</td>
    </tr>
    <tr>
        <th>tolerance</th>
        <td>The tolerance as given in optimizer_params.</td>
    </tr>
    <tr>
        <th>learning_rate_init</th>
        <td>The initial learning rate as given in optimizer_params.</td>
    </tr>
    <tr>
        <th>learning_rate_policy</th>
        <td>The learning rate policy as given in optimizer_params.</td>
    </tr>
    <tr>
        <th>momentum</th>
        <td>Momentum value as given in optimizer_params.</td>
    </tr>
    <tr>
        <th>nesterov</th>
        <td>Nesterov value as given in optimizer_params.</td>
    </tr>
    <tr>
        <th>n_iterations</th>
        <td>The number of iterations run.</td>
    </tr>
    <tr>
        <th>n_tries</th>
        <td>The number of tries as given in optimizer_params.</td>
    </tr>
    <tr>
        <th>layer_sizes</th>
        <td>The number of units in each layer including the input and output layers.</td>
    </tr>
    <tr>
        <th>activation</th>
        <td>The activation function.</td>
    </tr>
    <tr>
        <th>is_classification</th>
        <td>True if the model was trained for classification, False if it was trained
        for regression.</td>
    </tr>
    <tr>
        <th>classes</th>
        <td>The classes which were trained against (empty for regression).</td>
    </tr>
    <tr>
        <th>weights</th>
        <td>The weight column used during training for giving different
        weights to different rows.</td>
    </tr>
    <tr>
        <th>grouping_col</th>
        <td>NULL if no grouping_col was specified during training, and a
        comma-separated list of grouping column names if not.</td>
    </tr>

   </table>

A standardization table named \<output_table\>_standardization is also create,
that has the following columns:
  <table class="output">
    <tr>
        <th>mean</th>
        <td>The mean for all input features (used for normalization).</td>
    </tr>
    <tr>
        <th>std</th>
        <td>The standard deviation for all input features (used for normalization).</td>
    </tr>
    <tr>
        <th>grouping columns</th>
        <td>If grouping_col is specified during training, a column for each
        grouping column is created.</td>
    </tr>
  </table>

@anchor mlp_regression
@par Regression Training Function
The MLP regression training function has the following format:
<pre class="syntax">
mlp_regression(
    source_table,
    output_table,
    independent_varname,
    dependent_varname,
    hidden_layer_sizes,
    optimizer_params,
    activation,
    weights,
    warm_start,
    verbose,
    grouping_col
    )
</pre>

\b Arguments

Parameters for regression are largely the same as for classification. In the
model table, the loss refers to mean square error instead of cross entropy loss.
In the summary table, there is no classes column. The following
arguments have specifications which differ from mlp_classification:
<DL class="arglist">
<DT>dependent_varname</DT>
  <DD>TEXT. Name of the dependent variable column.
  For regression, supported types are any numeric type, or array
  of numeric types (for multiple regression).
  </DD>
</DL>

@anchor optimizer_params
@par Optimizer Parameters
Parameters in this section are supplied in the \e optimizer_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.

<pre class="syntax">
  'learning_rate_init = &lt;value>,
   learning_rate_policy = &lt;value>,
   gamma = &lt;value>,
   power = &lt;value>,
   iterations_per_step = &lt;value>,
   n_iterations = &lt;value>,
   n_tries = &lt;value>,
   lambda = &lt;value>,
   tolerance = &lt;value>,
   batch_size = &lt;value>,
   n_epochs = &lt;value>,
   momentum = &lt;value>,
   nesterov = &lt;value>'
</pre>
\b Optimizer \b Parameters
<DL class="arglist">

<DT>learning_rate_init</dt>
<DD>Default: 0.001.
Also known as the 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 tunes this parameter.
</DD>

<DT>learning_rate_policy</dt>
<DD>Default: constant.
One of 'constant', 'exp', 'inv' or 'step' or any prefix of these (e.g., 's' means 'step').
These are defined below, where 'iter' is the current iteration of SGD:
 - 'constant': learning_rate = learning_rate_init
 - 'exp': learning_rate = learning_rate_init * gamma^(iter)
 - 'inv': learning_rate = learning_rate_init * (iter+1)^(-power)
 - 'step': learning_rate = learning_rate_init * gamma^(floor(iter/iterations_per_step))
</DD>

<DT>gamma</dt>
<DD>Default: 0.1.
Decay rate for learning rate when learning_rate_policy is 'exp' or 'step'.
</DD>

<DT>power</dt>
<DD>Default: 0.5.
Exponent for learning_rate_policy = 'inv'.
</DD>

<DT>iterations_per_step</dt>
<DD>Default: 100.
Number of iterations to run before decreasing the learning rate by
a factor of gamma.  Valid for learning rate policy = 'step'.
</DD>

<DT>n_iterations</dt>
<DD>Default: 100. The maximum number of iterations allowed.
</DD>

<DT>n_tries</dt>
<DD>Default: 1. Number of times to retrain the network with randomly initialized
neural network weights.
</DD>

<DT>lambda</dt>
<DD>Default: 0. The regularization coefficient for L2 regularization.
</DD>

<DT>tolerance</dt>
<DD>Default: 0.001. The criterion to end iterations. The training stops whenever
the difference between the training models of two consecutive iterations is
smaller than \e tolerance or the iteration number is larger than \e n_iterations.
If you want to run the full number of iterations specified in \e n_interations,
set tolerance=0.0
</DD>

<DT>batch_size</dt>
<DD>Default: min(200, buffer_size) where buffer_size
is set in the mini-batch preprocessor.  The 'batch_size'
is the size of the mini-batch used in the optimizer.
This parameter is only used in the case of mini-batching.
</DD>

<DT>n_epochs</dt>
<DD>Default: 1. Represents the number of times
each batch is used by the optimizer.  This parameter
is only used in the case of mini-batching.
</DD>

<DT>momentum</dt>
<DD>Default: 0.9. Momentum can help accelerate learning and
avoid local minima when using gradient descent. Value must be in the
range 0 to 1, where 0 means no momentum.
</DD>

<DT>nesterov</dt>
<DD>Default: TRUE. Only used when the 'momentum' parameter is > 0.
Nesterov momentum can provide better results than using
classical momentum alone, due to its look-ahead characteristics. In classical
momentum we correct the velocity and then update the model with that velocity,
whereas in Nesterov Accelerated Gradient method, we first  move the model in the
direction of velocity, compute the gradient using this updated model, and then
add this gradient back into the model. The main difference being that in
classical momentum, we compute the gradient before updating the model whereas in
nesterov we first update the model and then compute the gradient from the
updated position.
</DD>

</DL>

@anchor predict
@par Prediction Function
Used to generate predictions on novel data given a previously trained model.
The same syntax is used for classification and regression.
<pre class="syntax">
mlp_predict(
    model_table,
    data_table,
    id_col_name,
    output_table,
    pred_type
    )
</pre>

\b Arguments
<DL class="arglist">
  <DT>model_table</DT>
  <DD>TEXT. Model table produced by the training function.</DD>

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

  <DT>id_col_name</DT>
  <DD>TEXT. The name of the id column in data_table.</DD>

  <DT>output_table</DT>
  <DD>TEXT. Name of the table where output predictions are written. If this
table name is already in use, an error is returned.  Table contains:</DD>
    <table class="output">
      <tr>
        <th>id</th>
        <td>Gives the 'id' for each prediction, corresponding to each row from the data_table.</td>
      </tr>
      <tr>
        <th>estimated_COL_NAME</th>
        <td>
        (For pred_type='response') The estimated class
         for classification or value for regression, where
         COL_NAME is the name of the column to be
         predicted from training data.
        </td>
      </tr>
      <tr>
        <th>prob_CLASS</th>
        <td>
        (For pred_type='prob' for classification) The
        probability of a given class CLASS as given by
        softmax. There will be one column for each class
        in the training data.
        </td>
      </tr>


  <DT>pred_type</DT>
  <DD>TEXT.  The type of output requested:
'response' gives the actual prediction,
'prob' gives the probability of each class.
For regression, only type='response' is defined.
</DD>
</DL>
</table>

@anchor example
@par Examples

<h4>Classification without Mini-Batching</h4>

-#  Create an input data set.
<pre class="example">
DROP TABLE IF EXISTS iris_data;
CREATE TABLE iris_data(
    id serial,
    attributes numeric[],
    class_text varchar,
    class integer,
    state varchar
);
INSERT INTO iris_data(id, attributes, class_text, class, state) VALUES
(1,ARRAY[5.0,3.2,1.2,0.2],'Iris_setosa',1,'Alaska'),
(2,ARRAY[5.5,3.5,1.3,0.2],'Iris_setosa',1,'Alaska'),
(3,ARRAY[4.9,3.1,1.5,0.1],'Iris_setosa',1,'Alaska'),
(4,ARRAY[4.4,3.0,1.3,0.2],'Iris_setosa',1,'Alaska'),
(5,ARRAY[5.1,3.4,1.5,0.2],'Iris_setosa',1,'Alaska'),
(6,ARRAY[5.0,3.5,1.3,0.3],'Iris_setosa',1,'Alaska'),
(7,ARRAY[4.5,2.3,1.3,0.3],'Iris_setosa',1,'Alaska'),
(8,ARRAY[4.4,3.2,1.3,0.2],'Iris_setosa',1,'Alaska'),
(9,ARRAY[5.0,3.5,1.6,0.6],'Iris_setosa',1,'Alaska'),
(10,ARRAY[5.1,3.8,1.9,0.4],'Iris_setosa',1,'Alaska'),
(11,ARRAY[4.8,3.0,1.4,0.3],'Iris_setosa',1,'Alaska'),
(12,ARRAY[5.1,3.8,1.6,0.2],'Iris_setosa',1,'Alaska'),
(13,ARRAY[5.7,2.8,4.5,1.3],'Iris_versicolor',2,'Alaska'),
(14,ARRAY[6.3,3.3,4.7,1.6],'Iris_versicolor',2,'Alaska'),
(15,ARRAY[4.9,2.4,3.3,1.0],'Iris_versicolor',2,'Alaska'),
(16,ARRAY[6.6,2.9,4.6,1.3],'Iris_versicolor',2,'Alaska'),
(17,ARRAY[5.2,2.7,3.9,1.4],'Iris_versicolor',2,'Alaska'),
(18,ARRAY[5.0,2.0,3.5,1.0],'Iris_versicolor',2,'Alaska'),
(19,ARRAY[5.9,3.0,4.2,1.5],'Iris_versicolor',2,'Alaska'),
(20,ARRAY[6.0,2.2,4.0,1.0],'Iris_versicolor',2,'Alaska'),
(21,ARRAY[6.1,2.9,4.7,1.4],'Iris_versicolor',2,'Alaska'),
(22,ARRAY[5.6,2.9,3.6,1.3],'Iris_versicolor',2,'Alaska'),
(23,ARRAY[6.7,3.1,4.4,1.4],'Iris_versicolor',2,'Alaska'),
(24,ARRAY[5.6,3.0,4.5,1.5],'Iris_versicolor',2,'Alaska'),
(25,ARRAY[5.8,2.7,4.1,1.0],'Iris_versicolor',2,'Alaska'),
(26,ARRAY[6.2,2.2,4.5,1.5],'Iris_versicolor',2,'Alaska'),
(27,ARRAY[5.6,2.5,3.9,1.1],'Iris_versicolor',2,'Alaska'),
(28,ARRAY[5.0,3.4,1.5,0.2],'Iris_setosa',1,'Tennessee'),
(29,ARRAY[4.4,2.9,1.4,0.2],'Iris_setosa',1,'Tennessee'),
(30,ARRAY[4.9,3.1,1.5,0.1],'Iris_setosa',1,'Tennessee'),
(31,ARRAY[5.4,3.7,1.5,0.2],'Iris_setosa',1,'Tennessee'),
(32,ARRAY[4.8,3.4,1.6,0.2],'Iris_setosa',1,'Tennessee'),
(33,ARRAY[4.8,3.0,1.4,0.1],'Iris_setosa',1,'Tennessee'),
(34,ARRAY[4.3,3.0,1.1,0.1],'Iris_setosa',1,'Tennessee'),
(35,ARRAY[5.8,4.0,1.2,0.2],'Iris_setosa',1,'Tennessee'),
(36,ARRAY[5.7,4.4,1.5,0.4],'Iris_setosa',1,'Tennessee'),
(37,ARRAY[5.4,3.9,1.3,0.4],'Iris_setosa',1,'Tennessee'),
(38,ARRAY[6.0,2.9,4.5,1.5],'Iris_versicolor',2,'Tennessee'),
(39,ARRAY[5.7,2.6,3.5,1.0],'Iris_versicolor',2,'Tennessee'),
(40,ARRAY[5.5,2.4,3.8,1.1],'Iris_versicolor',2,'Tennessee'),
(41,ARRAY[5.5,2.4,3.7,1.0],'Iris_versicolor',2,'Tennessee'),
(42,ARRAY[5.8,2.7,3.9,1.2],'Iris_versicolor',2,'Tennessee'),
(43,ARRAY[6.0,2.7,5.1,1.6],'Iris_versicolor',2,'Tennessee'),
(44,ARRAY[5.4,3.0,4.5,1.5],'Iris_versicolor',2,'Tennessee'),
(45,ARRAY[6.0,3.4,4.5,1.6],'Iris_versicolor',2,'Tennessee'),
(46,ARRAY[6.7,3.1,4.7,1.5],'Iris_versicolor',2,'Tennessee'),
(47,ARRAY[6.3,2.3,4.4,1.3],'Iris_versicolor',2,'Tennessee'),
(48,ARRAY[5.6,3.0,4.1,1.3],'Iris_versicolor',2,'Tennessee'),
(49,ARRAY[5.5,2.5,4.0,1.3],'Iris_versicolor',2,'Tennessee'),
(50,ARRAY[5.5,2.6,4.4,1.2],'Iris_versicolor',2,'Tennessee'),
(51,ARRAY[6.1,3.0,4.6,1.4],'Iris_versicolor',2,'Tennessee'),
(52,ARRAY[5.8,2.6,4.0,1.2],'Iris_versicolor',2,'Tennessee');
</pre>
-# Generate a multilayer perceptron with a single hidden layer of 5 units.
Use the attributes column as the independent variables, and use the class
column as the classification. Set the tolerance to 0 so that 500
iterations will be run. Use a hyperbolic tangent activation function.
The model will be written to mlp_model.
<pre class="example">
DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
SELECT madlib.mlp_classification(
    'iris_data',      -- Source table
    'mlp_model',      -- Destination table
    'attributes',     -- Input features
    'class_text',     -- Label
    ARRAY[5],         -- Number of units per layer
    'learning_rate_init=0.003,
    n_iterations=500,
    tolerance=0',     -- Optimizer params
    'tanh',           -- Activation function
    NULL,             -- Default weight (1)
    FALSE,            -- No warm start
    FALSE             -- Not verbose
);
</pre>
View the model:
<pre class="example">
\\x on
SELECT * FROM mlp_model;
</pre>
<pre class="result">
-[ RECORD 1 ]--+------------------------------------------------------------------------------------
coeff          | {-0.40378996718,0.0157490328855,-0.298904053444,-0.984152185093,-0.657684089715 ...
loss           | 0.0103518565103
num_iterations | 500
</pre>
View the model summary table:
<pre class="example">
SELECT * FROM mlp_model_summary;
</pre>
<pre class="result">
-[ RECORD 1 ]--------+------------------------------
source_table         | iris_data
independent_varname  | attributes
dependent_varname    | class_text
dependent_vartype    | character varying
tolerance            | 0
learning_rate_init   | 0.003
learning_rate_policy | constant
momentum             | 0.9
nesterov             | t
n_iterations         | 500
n_tries              | 1
layer_sizes          | {4,5,2}
activation           | tanh
is_classification    | t
classes              | {Iris_setosa,Iris_versicolor}
weights              | 1
grouping_col         | NULL
</pre>
View the model standardization table:
<pre class="example">
SELECT * FROM mlp_model_standardization;
</pre>
<pre class="result">
-[ RECORD 1 ]------------------------------------------------------------------
mean | {5.45961538461539,2.99807692307692,3.025,0.851923076923077}
std  | {0.598799958694505,0.498262513685689,1.41840579525043,0.550346179381454}
</pre>

-# Now let's use the model to predict. In the following example we will
use the training data set for prediction as well, which is not usual but serves to
show the syntax. The prediction is in the estimated_class_text column with the
actual value in the class_text column.
<pre class="example">
DROP TABLE IF EXISTS mlp_prediction;
\\x off
SELECT madlib.mlp_predict(
         'mlp_model',         -- Model table
         'iris_data',         -- Test data table
         'id',                -- Id column in test table
         'mlp_prediction',    -- Output table for predictions
         'response'           -- Output classes, not probabilities
     );
SELECT * FROM mlp_prediction JOIN iris_data USING (id) ORDER BY id;
</pre>
<pre class="result">
 id | estimated_class_text |    attributes     |   class_text    | class |   state
----+----------------------+-------------------+-----------------+-------+-----------
  1 | Iris_setosa          | {5.0,3.2,1.2,0.2} | Iris_setosa     |     1 | Alaska
  2 | Iris_setosa          | {5.5,3.5,1.3,0.2} | Iris_setosa     |     1 | Alaska
  3 | Iris_setosa          | {4.9,3.1,1.5,0.1} | Iris_setosa     |     1 | Alaska
  4 | Iris_setosa          | {4.4,3.0,1.3,0.2} | Iris_setosa     |     1 | Alaska
  5 | Iris_setosa          | {5.1,3.4,1.5,0.2} | Iris_setosa     |     1 | Alaska
  6 | Iris_setosa          | {5.0,3.5,1.3,0.3} | Iris_setosa     |     1 | Alaska
  7 | Iris_setosa          | {4.5,2.3,1.3,0.3} | Iris_setosa     |     1 | Alaska
  8 | Iris_setosa          | {4.4,3.2,1.3,0.2} | Iris_setosa     |     1 | Alaska
  9 | Iris_setosa          | {5.0,3.5,1.6,0.6} | Iris_setosa     |     1 | Alaska
 10 | Iris_setosa          | {5.1,3.8,1.9,0.4} | Iris_setosa     |     1 | Alaska
 11 | Iris_setosa          | {4.8,3.0,1.4,0.3} | Iris_setosa     |     1 | Alaska
 12 | Iris_setosa          | {5.1,3.8,1.6,0.2} | Iris_setosa     |     1 | Alaska
 13 | Iris_versicolor      | {5.7,2.8,4.5,1.3} | Iris_versicolor |     2 | Alaska
 14 | Iris_versicolor      | {6.3,3.3,4.7,1.6} | Iris_versicolor |     2 | Alaska
 15 | Iris_versicolor      | {4.9,2.4,3.3,1.0} | Iris_versicolor |     2 | Alaska
 16 | Iris_versicolor      | {6.6,2.9,4.6,1.3} | Iris_versicolor |     2 | Alaska
 17 | Iris_versicolor      | {5.2,2.7,3.9,1.4} | Iris_versicolor |     2 | Alaska
 18 | Iris_versicolor      | {5.0,2.0,3.5,1.0} | Iris_versicolor |     2 | Alaska
 19 | Iris_versicolor      | {5.9,3.0,4.2,1.5} | Iris_versicolor |     2 | Alaska
 20 | Iris_versicolor      | {6.0,2.2,4.0,1.0} | Iris_versicolor |     2 | Alaska
 21 | Iris_versicolor      | {6.1,2.9,4.7,1.4} | Iris_versicolor |     2 | Alaska
 22 | Iris_versicolor      | {5.6,2.9,3.6,1.3} | Iris_versicolor |     2 | Alaska
 23 | Iris_versicolor      | {6.7,3.1,4.4,1.4} | Iris_versicolor |     2 | Alaska
 24 | Iris_versicolor      | {5.6,3.0,4.5,1.5} | Iris_versicolor |     2 | Alaska
 25 | Iris_versicolor      | {5.8,2.7,4.1,1.0} | Iris_versicolor |     2 | Alaska
 26 | Iris_versicolor      | {6.2,2.2,4.5,1.5} | Iris_versicolor |     2 | Alaska
 27 | Iris_versicolor      | {5.6,2.5,3.9,1.1} | Iris_versicolor |     2 | Alaska
 28 | Iris_setosa          | {5.0,3.4,1.5,0.2} | Iris_setosa     |     1 | Tennessee
 29 | Iris_setosa          | {4.4,2.9,1.4,0.2} | Iris_setosa     |     1 | Tennessee
 30 | Iris_setosa          | {4.9,3.1,1.5,0.1} | Iris_setosa     |     1 | Tennessee
 31 | Iris_setosa          | {5.4,3.7,1.5,0.2} | Iris_setosa     |     1 | Tennessee
 32 | Iris_setosa          | {4.8,3.4,1.6,0.2} | Iris_setosa     |     1 | Tennessee
 33 | Iris_setosa          | {4.8,3.0,1.4,0.1} | Iris_setosa     |     1 | Tennessee
 34 | Iris_setosa          | {4.3,3.0,1.1,0.1} | Iris_setosa     |     1 | Tennessee
 35 | Iris_setosa          | {5.8,4.0,1.2,0.2} | Iris_setosa     |     1 | Tennessee
 36 | Iris_setosa          | {5.7,4.4,1.5,0.4} | Iris_setosa     |     1 | Tennessee
 37 | Iris_setosa          | {5.4,3.9,1.3,0.4} | Iris_setosa     |     1 | Tennessee
 38 | Iris_versicolor      | {6.0,2.9,4.5,1.5} | Iris_versicolor |     2 | Tennessee
 39 | Iris_versicolor      | {5.7,2.6,3.5,1.0} | Iris_versicolor |     2 | Tennessee
 40 | Iris_versicolor      | {5.5,2.4,3.8,1.1} | Iris_versicolor |     2 | Tennessee
 41 | Iris_versicolor      | {5.5,2.4,3.7,1.0} | Iris_versicolor |     2 | Tennessee
 42 | Iris_versicolor      | {5.8,2.7,3.9,1.2} | Iris_versicolor |     2 | Tennessee
 43 | Iris_versicolor      | {6.0,2.7,5.1,1.6} | Iris_versicolor |     2 | Tennessee
 44 | Iris_versicolor      | {5.4,3.0,4.5,1.5} | Iris_versicolor |     2 | Tennessee
 45 | Iris_versicolor      | {6.0,3.4,4.5,1.6} | Iris_versicolor |     2 | Tennessee
 46 | Iris_versicolor      | {6.7,3.1,4.7,1.5} | Iris_versicolor |     2 | Tennessee
 47 | Iris_versicolor      | {6.3,2.3,4.4,1.3} | Iris_versicolor |     2 | Tennessee
 48 | Iris_versicolor      | {5.6,3.0,4.1,1.3} | Iris_versicolor |     2 | Tennessee
 49 | Iris_versicolor      | {5.5,2.5,4.0,1.3} | Iris_versicolor |     2 | Tennessee
 50 | Iris_versicolor      | {5.5,2.6,4.4,1.2} | Iris_versicolor |     2 | Tennessee
 51 | Iris_versicolor      | {6.1,3.0,4.6,1.4} | Iris_versicolor |     2 | Tennessee
 52 | Iris_versicolor      | {5.8,2.6,4.0,1.2} | Iris_versicolor |     2 | Tennessee
(52 rows)
</pre>
Count the misclassifications:
<pre class="example">
SELECT COUNT(*) FROM mlp_prediction JOIN iris_data USING (id)
WHERE mlp_prediction.estimated_class_text != iris_data.class_text;
</pre>
<pre class="result">
 count
-------+
     0
</pre>

<h4>Classification with Mini-Batching</h4>

-#  Use the same data set as above.  Call mini-batch preprocessor:
<pre class="example">
DROP TABLE IF EXISTS iris_data_packed, iris_data_packed_summary, iris_data_packed_standardization;
SELECT madlib.minibatch_preprocessor('iris_data',         -- Source table
                                     'iris_data_packed',  -- Output table
                                     'class_text',        -- Dependent variable
                                     'attributes'        -- Independent variables
                                    );
</pre>
-# Train the classification model using similar parameters as before:
<pre class="example">
DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
SELECT madlib.mlp_classification(
    'iris_data_packed',      -- Output table from mini-batch preprocessor
    'mlp_model',             -- Destination table
    'independent_varname',   -- Hardcode to this, from table iris_data_packed
    'dependent_varname',     -- Hardcode to this, from table iris_data_packed
    ARRAY[5],                -- Number of units per layer
    'learning_rate_init=0.1,
    n_iterations=500,
    tolerance=0',            -- Optimizer params
    'tanh',                  -- Activation function
    NULL,                    -- Default weight (1)
    FALSE,                   -- No warm start
    FALSE                    -- Not verbose
);
</pre>
View the model:
<pre class="example">
\\x on
SELECT * FROM mlp_model;
</pre>
<pre class="result">
-[ RECORD 1 ]--+------------------------------------------------------------------------------------
coeff          | {-0.0780564661828377,-0.0781452670639994,0.3083605989842 ...
loss           | 0.00563534904146765
num_iterations | 500
</pre>

-# Now let's use the model to predict. As before we will
use the training data set for prediction as well, which is not usual but serves to
show the syntax. The prediction is in the estimated_class_text column with the
actual value in the class_text column.
<pre class="example">
DROP TABLE IF EXISTS mlp_prediction;
\\x off
SELECT madlib.mlp_predict(
         'mlp_model',         -- Model table
         'iris_data',         -- Test data table
         'id',                -- Id column in test table
         'mlp_prediction',    -- Output table for predictions
         'response'           -- Output classes, not probabilities
     );
SELECT * FROM mlp_prediction JOIN iris_data USING (id) ORDER BY id;
</pre>
<pre class="result">
 id | estimated_class_text |    attributes     |   class_text    | class |   state
----+----------------------+-------------------+-----------------+-------+-----------
  1 | Iris_setosa          | {5.0,3.2,1.2,0.2} | Iris_setosa     |     1 | Alaska
  2 | Iris_setosa          | {5.5,3.5,1.3,0.2} | Iris_setosa     |     1 | Alaska
  3 | Iris_setosa          | {4.9,3.1,1.5,0.1} | Iris_setosa     |     1 | Alaska
  4 | Iris_setosa          | {4.4,3.0,1.3,0.2} | Iris_setosa     |     1 | Alaska
  5 | Iris_setosa          | {5.1,3.4,1.5,0.2} | Iris_setosa     |     1 | Alaska
  6 | Iris_setosa          | {5.0,3.5,1.3,0.3} | Iris_setosa     |     1 | Alaska
  7 | Iris_setosa          | {4.5,2.3,1.3,0.3} | Iris_setosa     |     1 | Alaska
  8 | Iris_setosa          | {4.4,3.2,1.3,0.2} | Iris_setosa     |     1 | Alaska
  9 | Iris_setosa          | {5.0,3.5,1.6,0.6} | Iris_setosa     |     1 | Alaska
 10 | Iris_setosa          | {5.1,3.8,1.9,0.4} | Iris_setosa     |     1 | Alaska
 11 | Iris_setosa          | {4.8,3.0,1.4,0.3} | Iris_setosa     |     1 | Alaska
 12 | Iris_setosa          | {5.1,3.8,1.6,0.2} | Iris_setosa     |     1 | Alaska
 13 | Iris_versicolor      | {5.7,2.8,4.5,1.3} | Iris_versicolor |     2 | Alaska
 14 | Iris_versicolor      | {6.3,3.3,4.7,1.6} | Iris_versicolor |     2 | Alaska
 15 | Iris_versicolor      | {4.9,2.4,3.3,1.0} | Iris_versicolor |     2 | Alaska
 16 | Iris_versicolor      | {6.6,2.9,4.6,1.3} | Iris_versicolor |     2 | Alaska
 17 | Iris_versicolor      | {5.2,2.7,3.9,1.4} | Iris_versicolor |     2 | Alaska
 18 | Iris_versicolor      | {5.0,2.0,3.5,1.0} | Iris_versicolor |     2 | Alaska
 19 | Iris_versicolor      | {5.9,3.0,4.2,1.5} | Iris_versicolor |     2 | Alaska
 20 | Iris_versicolor      | {6.0,2.2,4.0,1.0} | Iris_versicolor |     2 | Alaska
 21 | Iris_versicolor      | {6.1,2.9,4.7,1.4} | Iris_versicolor |     2 | Alaska
 22 | Iris_versicolor      | {5.6,2.9,3.6,1.3} | Iris_versicolor |     2 | Alaska
 23 | Iris_versicolor      | {6.7,3.1,4.4,1.4} | Iris_versicolor |     2 | Alaska
 24 | Iris_versicolor      | {5.6,3.0,4.5,1.5} | Iris_versicolor |     2 | Alaska
 25 | Iris_versicolor      | {5.8,2.7,4.1,1.0} | Iris_versicolor |     2 | Alaska
 26 | Iris_versicolor      | {6.2,2.2,4.5,1.5} | Iris_versicolor |     2 | Alaska
 27 | Iris_versicolor      | {5.6,2.5,3.9,1.1} | Iris_versicolor |     2 | Alaska
 28 | Iris_setosa          | {5.0,3.4,1.5,0.2} | Iris_setosa     |     1 | Tennessee
 29 | Iris_setosa          | {4.4,2.9,1.4,0.2} | Iris_setosa     |     1 | Tennessee
 30 | Iris_setosa          | {4.9,3.1,1.5,0.1} | Iris_setosa     |     1 | Tennessee
 31 | Iris_setosa          | {5.4,3.7,1.5,0.2} | Iris_setosa     |     1 | Tennessee
 32 | Iris_setosa          | {4.8,3.4,1.6,0.2} | Iris_setosa     |     1 | Tennessee
 33 | Iris_setosa          | {4.8,3.0,1.4,0.1} | Iris_setosa     |     1 | Tennessee
 34 | Iris_setosa          | {4.3,3.0,1.1,0.1} | Iris_setosa     |     1 | Tennessee
 35 | Iris_setosa          | {5.8,4.0,1.2,0.2} | Iris_setosa     |     1 | Tennessee
 36 | Iris_setosa          | {5.7,4.4,1.5,0.4} | Iris_setosa     |     1 | Tennessee
 37 | Iris_setosa          | {5.4,3.9,1.3,0.4} | Iris_setosa     |     1 | Tennessee
 38 | Iris_versicolor      | {6.0,2.9,4.5,1.5} | Iris_versicolor |     2 | Tennessee
 39 | Iris_versicolor      | {5.7,2.6,3.5,1.0} | Iris_versicolor |     2 | Tennessee
 40 | Iris_versicolor      | {5.5,2.4,3.8,1.1} | Iris_versicolor |     2 | Tennessee
 41 | Iris_versicolor      | {5.5,2.4,3.7,1.0} | Iris_versicolor |     2 | Tennessee
 42 | Iris_versicolor      | {5.8,2.7,3.9,1.2} | Iris_versicolor |     2 | Tennessee
 43 | Iris_versicolor      | {6.0,2.7,5.1,1.6} | Iris_versicolor |     2 | Tennessee
 44 | Iris_versicolor      | {5.4,3.0,4.5,1.5} | Iris_versicolor |     2 | Tennessee
 45 | Iris_versicolor      | {6.0,3.4,4.5,1.6} | Iris_versicolor |     2 | Tennessee
 46 | Iris_versicolor      | {6.7,3.1,4.7,1.5} | Iris_versicolor |     2 | Tennessee
 47 | Iris_versicolor      | {6.3,2.3,4.4,1.3} | Iris_versicolor |     2 | Tennessee
 48 | Iris_versicolor      | {5.6,3.0,4.1,1.3} | Iris_versicolor |     2 | Tennessee
 49 | Iris_versicolor      | {5.5,2.5,4.0,1.3} | Iris_versicolor |     2 | Tennessee
 50 | Iris_versicolor      | {5.5,2.6,4.4,1.2} | Iris_versicolor |     2 | Tennessee
 51 | Iris_versicolor      | {6.1,3.0,4.6,1.4} | Iris_versicolor |     2 | Tennessee
 52 | Iris_versicolor      | {5.8,2.6,4.0,1.2} | Iris_versicolor |     2 | Tennessee
(52 rows)
</pre>
Count the misclassifications:
<pre class="example">
SELECT COUNT(*) FROM mlp_prediction JOIN iris_data USING (id)
WHERE mlp_prediction.estimated_class_text != iris_data.class_text;
</pre>
<pre class="result">
 count
-------+
     0
</pre>

<h4>Classification with Other Parameters</h4>

-# Now, use the n_tries optimizer parameter to learn and choose the best model
among n_tries number of models learnt by the algorithm. Run only for 50 iterations
and choose the best model from this short run. Note we are not using mini-batching
here.
<pre class="example">
DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
SELECT madlib.mlp_classification(
    'iris_data',      -- Source table
    'mlp_model',      -- Destination table
    'attributes',     -- Input features
    'class_text',     -- Label
    ARRAY[5],         -- Number of units per layer
    'learning_rate_init=0.003,
    n_iterations=50,
    tolerance=0,
    n_tries=3',       -- Optimizer params, with n_tries
    'tanh',           -- Activation function
    NULL,             -- Default weight (1)
    FALSE,            -- No warm start
    FALSE             -- Not verbose
);
</pre>
View the model:
<pre class="example">
\\x on
SELECT * FROM mlp_model;
</pre>
<pre class="result">
-[ RECORD 1 ]--+------------------------------------------------------------------------------------
coeff          | {0.000156316559088915,0.131131017223563,-0.293990512682215 ...
loss           | 0.142238768280717
num_iterations | 50
</pre>

-# Next, use the warm_start parameter to start learning a new model, using
the coefficients already present in mlp_model. Note that we must not drop
the mlp_model table, and cannot use the n_tries parameter if warm_start is
used.
<pre class="example">
SELECT madlib.mlp_classification(
    'iris_data',      -- Source table
    'mlp_model',      -- Destination table
    'attributes',     -- Input features
    'class_text',     -- Label
    ARRAY[5],         -- Number of units per layer
    'learning_rate_init=0.003,
    n_iterations=450,
    tolerance=0',     -- Optimizer params
    'tanh',           -- Activation function
    NULL,             -- Default weight (1)
    TRUE,             -- Warm start
    FALSE             -- Not verbose
);
</pre>
View the model:
<pre class="example">
\\x on
SELECT * FROM mlp_model;
</pre>
<pre class="result">
-[ RECORD 1 ]--+------------------------------------------------------------------------------------
coeff          | {0.0883013960215441,0.235944854050211,-0.398126039487036 ...
loss           | 0.00818899646775659
num_iterations | 450
</pre>
Notice that the loss is lower compared to the previous example, despite
having the same values for every other parameter. This is because the algorithm
learned three different models starting with a different set of initial weights
for the coefficients, and chose the best model among them as the initial
weights for the coefficients when run with warm start.

<h4>Classification with Grouping</h4>

-# Next, group the training data by state, and learn a different model for each state.
Note we are not using mini-batching in this example.
<pre class="example">
DROP TABLE IF EXISTS mlp_model_group, mlp_model_group_summary, mlp_model_group_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
SELECT madlib.mlp_classification(
    'iris_data',        -- Source table
    'mlp_model_group',  -- Destination table
    'attributes',       -- Input features
    'class_text',       -- Label
    ARRAY[5],           -- Number of units per layer
    'learning_rate_init=0.003,
    n_iterations=500,   -- Optimizer params
    tolerance=0',
    'tanh',             -- Activation function
    NULL,               -- Default weight (1)
    FALSE,              -- No warm start
    FALSE,              -- Not verbose
    'state'             -- Grouping column
);
</pre>
View the model:
<pre class="example">
\\x on
SELECT * FROM mlp_model_group ORDER BY state;
</pre>
<pre class="result">
-[ RECORD 1 ]--+------------------------------------------------------------------------------------
state          | Alaska
coeff          | {-0.51246602223,-0.78952457411,0.454192045225,0.223214894458,0.188804700547 ...
loss           | 0.0225081995679
num_iterations | 500
-[ RECORD 2 ]--+------------------------------------------------------------------------------------
state          | Tennessee
coeff          | {-0.215009937565,0.116581594162,-0.397643279185,0.919193295184,-0.0811341736111 ...
loss           | 0.0182854983946
num_iterations | 500
</pre>
A separate model is learnt for each state, and the result table displays the name of
the state (grouping column) associated with the model.

-# Prediction based on grouping using the state column:
<pre class="example">
DROP TABLE IF EXISTS mlp_prediction;
SELECT madlib.mlp_predict(
         'mlp_model_group',   -- Model table
         'iris_data',         -- Test data table
         'id',                -- Id column in test table
         'mlp_prediction',    -- Output table for predictions
         'response'           -- Output classes, not probabilities
     );
SELECT * FROM mlp_prediction JOIN iris_data USING (state,id) ORDER BY state, id;
</pre>
Result for the classification model:
<pre class="result">
   state   | id | estimated_class_text |    attributes     |   class_text    | class
-----------+----+----------------------+-------------------+-----------------+-------
 Alaska    |  1 | Iris_setosa          | {5.0,3.2,1.2,0.2} | Iris_setosa     |     1
 Alaska    |  2 | Iris_setosa          | {5.5,3.5,1.3,0.2} | Iris_setosa     |     1
 Alaska    |  3 | Iris_setosa          | {4.9,3.1,1.5,0.1} | Iris_setosa     |     1
 Alaska    |  4 | Iris_setosa          | {4.4,3.0,1.3,0.2} | Iris_setosa     |     1
 Alaska    |  5 | Iris_setosa          | {5.1,3.4,1.5,0.2} | Iris_setosa     |     1
 Alaska    |  6 | Iris_setosa          | {5.0,3.5,1.3,0.3} | Iris_setosa     |     1
 Alaska    |  7 | Iris_setosa          | {4.5,2.3,1.3,0.3} | Iris_setosa     |     1
 Alaska    |  8 | Iris_setosa          | {4.4,3.2,1.3,0.2} | Iris_setosa     |     1
 Alaska    |  9 | Iris_setosa          | {5.0,3.5,1.6,0.6} | Iris_setosa     |     1
 Alaska    | 10 | Iris_setosa          | {5.1,3.8,1.9,0.4} | Iris_setosa     |     1
 Alaska    | 11 | Iris_setosa          | {4.8,3.0,1.4,0.3} | Iris_setosa     |     1
 Alaska    | 12 | Iris_setosa          | {5.1,3.8,1.6,0.2} | Iris_setosa     |     1
 Alaska    | 13 | Iris_versicolor      | {5.7,2.8,4.5,1.3} | Iris_versicolor |     2
 Alaska    | 14 | Iris_versicolor      | {6.3,3.3,4.7,1.6} | Iris_versicolor |     2
 Alaska    | 15 | Iris_versicolor      | {4.9,2.4,3.3,1.0} | Iris_versicolor |     2
 Alaska    | 16 | Iris_versicolor      | {6.6,2.9,4.6,1.3} | Iris_versicolor |     2
 Alaska    | 17 | Iris_versicolor      | {5.2,2.7,3.9,1.4} | Iris_versicolor |     2
 Alaska    | 18 | Iris_versicolor      | {5.0,2.0,3.5,1.0} | Iris_versicolor |     2
 Alaska    | 19 | Iris_versicolor      | {5.9,3.0,4.2,1.5} | Iris_versicolor |     2
 Alaska    | 20 | Iris_versicolor      | {6.0,2.2,4.0,1.0} | Iris_versicolor |     2
 Alaska    | 21 | Iris_versicolor      | {6.1,2.9,4.7,1.4} | Iris_versicolor |     2
 Alaska    | 22 | Iris_versicolor      | {5.6,2.9,3.6,1.3} | Iris_versicolor |     2
 Alaska    | 23 | Iris_versicolor      | {6.7,3.1,4.4,1.4} | Iris_versicolor |     2
 Alaska    | 24 | Iris_versicolor      | {5.6,3.0,4.5,1.5} | Iris_versicolor |     2
 Alaska    | 25 | Iris_versicolor      | {5.8,2.7,4.1,1.0} | Iris_versicolor |     2
 Alaska    | 26 | Iris_versicolor      | {6.2,2.2,4.5,1.5} | Iris_versicolor |     2
 Alaska    | 27 | Iris_versicolor      | {5.6,2.5,3.9,1.1} | Iris_versicolor |     2
 Tennessee | 28 | Iris_setosa          | {5.0,3.4,1.5,0.2} | Iris_setosa     |     1
 Tennessee | 29 | Iris_setosa          | {4.4,2.9,1.4,0.2} | Iris_setosa     |     1
 Tennessee | 30 | Iris_setosa          | {4.9,3.1,1.5,0.1} | Iris_setosa     |     1
 Tennessee | 31 | Iris_setosa          | {5.4,3.7,1.5,0.2} | Iris_setosa     |     1
 Tennessee | 32 | Iris_setosa          | {4.8,3.4,1.6,0.2} | Iris_setosa     |     1
 Tennessee | 33 | Iris_setosa          | {4.8,3.0,1.4,0.1} | Iris_setosa     |     1
 Tennessee | 34 | Iris_setosa          | {4.3,3.0,1.1,0.1} | Iris_setosa     |     1
 Tennessee | 35 | Iris_setosa          | {5.8,4.0,1.2,0.2} | Iris_setosa     |     1
 Tennessee | 36 | Iris_setosa          | {5.7,4.4,1.5,0.4} | Iris_setosa     |     1
 Tennessee | 37 | Iris_setosa          | {5.4,3.9,1.3,0.4} | Iris_setosa     |     1
 Tennessee | 38 | Iris_versicolor      | {6.0,2.9,4.5,1.5} | Iris_versicolor |     2
 Tennessee | 39 | Iris_versicolor      | {5.7,2.6,3.5,1.0} | Iris_versicolor |     2
 Tennessee | 40 | Iris_versicolor      | {5.5,2.4,3.8,1.1} | Iris_versicolor |     2
 Tennessee | 41 | Iris_versicolor      | {5.5,2.4,3.7,1.0} | Iris_versicolor |     2
 Tennessee | 42 | Iris_versicolor      | {5.8,2.7,3.9,1.2} | Iris_versicolor |     2
 Tennessee | 43 | Iris_versicolor      | {6.0,2.7,5.1,1.6} | Iris_versicolor |     2
 Tennessee | 44 | Iris_versicolor      | {5.4,3.0,4.5,1.5} | Iris_versicolor |     2
 Tennessee | 45 | Iris_versicolor      | {6.0,3.4,4.5,1.6} | Iris_versicolor |     2
 Tennessee | 46 | Iris_versicolor      | {6.7,3.1,4.7,1.5} | Iris_versicolor |     2
 Tennessee | 47 | Iris_versicolor      | {6.3,2.3,4.4,1.3} | Iris_versicolor |     2
 Tennessee | 48 | Iris_versicolor      | {5.6,3.0,4.1,1.3} | Iris_versicolor |     2
 Tennessee | 49 | Iris_versicolor      | {5.5,2.5,4.0,1.3} | Iris_versicolor |     2
 Tennessee | 50 | Iris_versicolor      | {5.5,2.6,4.4,1.2} | Iris_versicolor |     2
 Tennessee | 51 | Iris_versicolor      | {6.1,3.0,4.6,1.4} | Iris_versicolor |     2
 Tennessee | 52 | Iris_versicolor      | {5.8,2.6,4.0,1.2} | Iris_versicolor |     2
(52 rows)
</pre>

<h4>Regression without Mini-Batching</h4>

-# Create a dataset with housing prices data.
<pre class="example">
DROP TABLE IF EXISTS lin_housing;
CREATE TABLE lin_housing (id serial, x numeric[], zipcode int, y float8);
INSERT INTO lin_housing(id, x, zipcode, y) VALUES
(1,ARRAY[1,0.00632,18.00,2.310,0,0.5380,6.5750,65.20,4.0900,1,296.0,15.30,396.90,4.98],94016,24.00),
(2,ARRAY[1,0.02731,0.00,7.070,0,0.4690,6.4210,78.90,4.9671,2,242.0,17.80,396.90,9.14],94016,21.60),
(3,ARRAY[1,0.02729,0.00,7.070,0,0.4690,7.1850,61.10,4.9671,2,242.0,17.80,392.83,4.03],94016,34.70),
(4,ARRAY[1,0.03237,0.00,2.180,0,0.4580,6.9980,45.80,6.0622,3,222.0,18.70,394.63,2.94],94016,33.40),
(5,ARRAY[1,0.06905,0.00,2.180,0,0.4580,7.1470,54.20,6.0622,3,222.0,18.70,396.90,5.33],94016,36.20),
(6,ARRAY[1,0.02985,0.00,2.180,0,0.4580,6.4300,58.70,6.0622,3,222.0,18.70,394.12,5.21],94016,28.70),
(7,ARRAY[1,0.08829,12.50,7.870,0,0.5240,6.0120,66.60,5.5605,5,311.0,15.20,395.60,12.43],94016,22.90),
(8,ARRAY[1,0.14455,12.50,7.870,0,0.5240,6.1720,96.10,5.9505,5,311.0,15.20,396.90,19.15],94016,27.10),
(9,ARRAY[1,0.21124,12.50,7.870,0,0.5240,5.6310,100.00,6.0821,5,311.0,15.20,386.63,29.93],94016,16.50),
(10,ARRAY[1,0.17004,12.50,7.870,0,0.5240,6.0040,85.90,6.5921,5,311.0,15.20,386.71,17.10],94016,18.90),
(11,ARRAY[1,0.22489,12.50,7.870,0,0.5240,6.3770,94.30,6.3467,5,311.0,15.20,392.52,20.45],94016,15.00),
(12,ARRAY[1,0.11747,12.50,7.870,0,0.5240,6.0090,82.90,6.2267,5,311.0,15.20,396.90,13.27],20001,18.90),
(13,ARRAY[1,0.09378,12.50,7.870,0,0.5240,5.8890,39.00,5.4509,5,311.0,15.20,390.50,15.71],20001,21.70),
(14,ARRAY[1,0.62976,0.00,8.140,0,0.5380,5.9490,61.80,4.7075,4,307.0,21.00,396.90,8.26],20001,20.40),
(15,ARRAY[1,0.63796,0.00,8.140,0,0.5380,6.0960,84.50,4.4619,4,307.0,21.00,380.02,10.26],20001,18.20),
(16,ARRAY[1,0.62739,0.00,8.140,0,0.5380,5.8340,56.50,4.4986,4,307.0,21.00,395.62,8.47],20001,19.90),
(17,ARRAY[1,1.05393,0.00,8.140,0,0.5380,5.9350,29.30,4.4986,4,307.0,21.00,386.85,6.58],20001, 23.10),
(18,ARRAY[1,0.78420,0.00,8.140,0,0.5380,5.9900,81.70,4.2579,4,307.0,21.00,386.75,14.67],20001,17.50),
(19,ARRAY[1,0.80271,0.00,8.140,0,0.5380,5.4560,36.60,3.7965,4,307.0,21.00,288.99,11.69],20001,20.20),
(20,ARRAY[1,0.72580,0.00,8.140,0,0.5380,5.7270,69.50,3.7965,4,307.0,21.00,390.95,11.28],20001,18.20);
</pre>

-# Now train a regression model using a multilayer
perceptron with two hidden layers of twenty five nodes each:
<pre class="example">
DROP TABLE IF EXISTS mlp_regress, mlp_regress_summary, mlp_regress_standardization;
SELECT setseed(0);
SELECT madlib.mlp_regression(
    'lin_housing',    -- Source table
    'mlp_regress',    -- Desination table
    'x',              -- Input features
    'y',              -- Dependent variable
    ARRAY[25,25],     -- Number of units per layer
    'learning_rate_init=0.001,
    n_iterations=500,
    lambda=0.001,
    tolerance=0',     -- Optimizer params
    'relu',           -- Activation function
    NULL,             -- Default weight (1)
    FALSE,            -- No warm start
    FALSE             -- Not verbose
);
</pre>
View the model:
<pre class="example">
\\x on
SELECT * FROM mlp_regress;
</pre>
<pre class="result">
[ RECORD 1 ]--+-------------------------------------------------------------------------------------
coeff          | {-0.250057620174,0.0630805938982,-0.290635490112,-0.382966162592,-0.212206338909...
loss           | 1.07042781236
num_iterations | 500
</pre>

-# Prediction using the regression model:
<pre class="example">
DROP TABLE IF EXISTS mlp_regress_prediction;
SELECT madlib.mlp_predict(
         'mlp_regress',               -- Model table
         'lin_housing',               -- Test data table
         'id',                        -- Id column in test table
         'mlp_regress_prediction',    -- Output table for predictions
         'response'                   -- Output values, not probabilities
     );
</pre>
View results:
<pre class="example">
SELECT * FROM lin_housing JOIN mlp_regress_prediction USING (id) ORDER BY id;
</pre>
<pre class="result">
 id |                                         x                                        | zipcode |  y   |   estimated_y
----+----------------------------------------------------------------------------------+---------+------+------------------
  1 | {1,0.00632,18.00,2.310,0,0.5380,6.5750,65.20,4.0900,1,296.0,15.30,396.90,4.98}   |   94016 |   24 | 23.9989087488259
  2 | {1,0.02731,0.00,7.070,0,0.4690,6.4210,78.90,4.9671,2,242.0,17.80,396.90,9.14}    |   94016 | 21.6 | 21.5983177932005
  3 | {1,0.02729,0.00,7.070,0,0.4690,7.1850,61.10,4.9671,2,242.0,17.80,392.83,4.03}    |   94016 | 34.7 | 34.7102398021623
  4 | {1,0.03237,0.00,2.180,0,0.4580,6.9980,45.80,6.0622,3,222.0,18.70,394.63,2.94}    |   94016 | 33.4 | 33.4221257351015
  5 | {1,0.06905,0.00,2.180,0,0.4580,7.1470,54.20,6.0622,3,222.0,18.70,396.90,5.33}    |   94016 | 36.2 | 36.1523886001663
  6 | {1,0.02985,0.00,2.180,0,0.4580,6.4300,58.70,6.0622,3,222.0,18.70,394.12,5.21}    |   94016 | 28.7 |  28.723894783928
  7 | {1,0.08829,12.50,7.870,0,0.5240,6.0120,66.60,5.5605,5,311.0,15.20,395.60,12.43}  |   94016 | 22.9 | 22.6515242795835
  8 | {1,0.14455,12.50,7.870,0,0.5240,6.1720,96.10,5.9505,5,311.0,15.20,396.90,19.15}  |   94016 | 27.1 | 25.7615314879354
  9 | {1,0.21124,12.50,7.870,0,0.5240,5.6310,100.00,6.0821,5,311.0,15.20,386.63,29.93} |   94016 | 16.5 | 15.7368298351732
 10 | {1,0.17004,12.50,7.870,0,0.5240,6.0040,85.90,6.5921,5,311.0,15.20,386.71,17.10}  |   94016 | 18.9 | 16.8850496141437
 11 | {1,0.22489,12.50,7.870,0,0.5240,6.3770,94.30,6.3467,5,311.0,15.20,392.52,20.45}  |   94016 |   15 | 14.9150416339458
 12 | {1,0.11747,12.50,7.870,0,0.5240,6.0090,82.90,6.2267,5,311.0,15.20,396.90,13.27}  |   20001 | 18.9 | 19.4541629864106
 13 | {1,0.09378,12.50,7.870,0,0.5240,5.8890,39.00,5.4509,5,311.0,15.20,390.50,15.71}  |   20001 | 21.7 |  21.715554997762
 14 | {1,0.62976,0.00,8.140,0,0.5380,5.9490,61.80,4.7075,4,307.0,21.00,396.90,8.26}    |   20001 | 20.4 | 20.3181247234996
 15 | {1,0.63796,0.00,8.140,0,0.5380,6.0960,84.50,4.4619,4,307.0,21.00,380.02,10.26}   |   20001 | 18.2 | 18.5026399122209
 16 | {1,0.62739,0.00,8.140,0,0.5380,5.8340,56.50,4.4986,4,307.0,21.00,395.62,8.47}    |   20001 | 19.9 | 19.9131696333521
 17 | {1,1.05393,0.00,8.140,0,0.5380,5.9350,29.30,4.4986,4,307.0,21.00,386.85,6.58}    |   20001 | 23.1 | 23.1757650468106
 18 | {1,0.78420,0.00,8.140,0,0.5380,5.9900,81.70,4.2579,4,307.0,21.00,386.75,14.67}   |   20001 | 17.5 | 17.2671872543377
 19 | {1,0.80271,0.00,8.140,0,0.5380,5.4560,36.60,3.7965,4,307.0,21.00,288.99,11.69}   |   20001 | 20.2 | 20.1073474558796
 20 | {1,0.72580,0.00,8.140,0,0.5380,5.7270,69.50,3.7965,4,307.0,21.00,390.95,11.28}   |   20001 | 18.2 | 18.2143446340975
(20 rows)
</pre>
RMS error:
<pre class="example">
SELECT SQRT(AVG((y-estimated_y)*(y-estimated_y))) as rms_error FROM lin_housing
JOIN mlp_regress_prediction USING (id);
</pre>
<pre class="result">
    rms_error
------------------+
 0.544960829104004
</pre>

<h4>Regression with Mini-Batching</h4>

-# Call min-batch preprocessor using
the same data set as above:
<pre class="example">
DROP TABLE IF EXISTS lin_housing_packed, lin_housing_packed_summary, lin_housing_packed_standardization;
SELECT madlib.minibatch_preprocessor('lin_housing',         -- Source table
                                     'lin_housing_packed',  -- Output table
                                     'y',                   -- Dependent variable
                                     'x'                   -- Independent variables
                                     );
</pre>
-# Train regression model with mini-batching
<pre class="example">
DROP TABLE IF EXISTS mlp_regress, mlp_regress_summary, mlp_regress_standardization;
SELECT setseed(0);
SELECT madlib.mlp_regression(
    'lin_housing_packed',    -- Source table
    'mlp_regress',           -- Desination table
    'independent_varname',   -- Hardcode to this, from table lin_housing_packed
    'dependent_varname',     -- Hardcode to this, from table lin_housing_packed
    ARRAY[25,25],            -- Number of units per layer
    'learning_rate_init=0.01,
    n_iterations=500,
    lambda=0.001,
    tolerance=0',            -- Optimizer params
    'tanh',                  -- Activation function
    NULL,                    -- Default weight (1)
    FALSE,                   -- No warm start
    FALSE                    -- Not verbose
);
</pre>
View model:
<pre class="example">
\\x on
SELECT * FROM mlp_regress;
</pre>
<pre class="result">
-[ RECORD 1 ]--+-------------------------------------------------------------
coeff          | {0.0395865908810001,-0.164860448878703,-0.132787863194324...
loss           | 0.0442383714892138
num_iterations | 500
</pre>
-# Prediction for regression:
<pre class="example">
DROP TABLE IF EXISTS mlp_regress_prediction;
SELECT madlib.mlp_predict(
         'mlp_regress',               -- Model table
         'lin_housing',               -- Test data table
         'id',                        -- Id column in test table
         'mlp_regress_prediction',    -- Output table for predictions
         'response'                   -- Output values, not probabilities
     );
\\x off
SELECT *, ABS(y-estimated_y) as abs_diff FROM lin_housing JOIN mlp_regress_prediction USING (id) ORDER BY id;
</pre>
<pre class="result">
 id |                                        x                                         | zipcode |  y   | zipcode |   estimated_y    |      abs_diff
----+----------------------------------------------------------------------------------+---------+------+---------+------------------+--------------------
  1 | {1,0.00632,18.00,2.310,0,0.5380,6.5750,65.20,4.0900,1,296.0,15.30,396.90,4.98}   |   94016 |   24 |   94016 | 23.9714991250013 | 0.0285008749987092
  2 | {1,0.02731,0.00,7.070,0,0.4690,6.4210,78.90,4.9671,2,242.0,17.80,396.90,9.14}    |   94016 | 21.6 |   94016 | 22.3655180133895 |  0.765518013389535
  3 | {1,0.02729,0.00,7.070,0,0.4690,7.1850,61.10,4.9671,2,242.0,17.80,392.83,4.03}    |   94016 | 34.7 |   94016 | 33.8620767428645 |  0.837923257135465
  4 | {1,0.03237,0.00,2.180,0,0.4580,6.9980,45.80,6.0622,3,222.0,18.70,394.63,2.94}    |   94016 | 33.4 |   94016 | 35.3094157686524 |   1.90941576865244
  5 | {1,0.06905,0.00,2.180,0,0.4580,7.1470,54.20,6.0622,3,222.0,18.70,396.90,5.33}    |   94016 | 36.2 |   94016 | 35.0379122731818 |   1.16208772681817
  6 | {1,0.02985,0.00,2.180,0,0.4580,6.4300,58.70,6.0622,3,222.0,18.70,394.12,5.21}    |   94016 | 28.7 |   94016 | 27.5207943492151 |   1.17920565078487
  7 | {1,0.08829,12.50,7.870,0,0.5240,6.0120,66.60,5.5605,5,311.0,15.20,395.60,12.43}  |   94016 | 22.9 |   94016 | 24.9841422781166 |    2.0841422781166
  8 | {1,0.14455,12.50,7.870,0,0.5240,6.1720,96.10,5.9505,5,311.0,15.20,396.90,19.15}  |   94016 | 27.1 |   94016 | 24.5403994064793 |   2.55960059352067
  9 | {1,0.21124,12.50,7.870,0,0.5240,5.6310,100.00,6.0821,5,311.0,15.20,386.63,29.93} |   94016 | 16.5 |   94016 | 17.2588278443879 |   0.75882784438787
 10 | {1,0.17004,12.50,7.870,0,0.5240,6.0040,85.90,6.5921,5,311.0,15.20,386.71,17.10}  |   94016 | 18.9 |   94016 | 17.0600407532569 |    1.8399592467431
 11 | {1,0.22489,12.50,7.870,0,0.5240,6.3770,94.30,6.3467,5,311.0,15.20,392.52,20.45}  |   94016 |   15 |   94016 | 15.2284207930287 |  0.228420793028732
 12 | {1,0.11747,12.50,7.870,0,0.5240,6.0090,82.90,6.2267,5,311.0,15.20,396.90,13.27}  |   20001 | 18.9 |   20001 | 19.2272848285357 |  0.327284828535671
 13 | {1,0.09378,12.50,7.870,0,0.5240,5.8890,39.00,5.4509,5,311.0,15.20,390.50,15.71}  |   20001 | 21.7 |   20001 | 21.3979318641202 |  0.302068135879811
 14 | {1,0.62976,0.00,8.140,0,0.5380,5.9490,61.80,4.7075,4,307.0,21.00,396.90,8.26}    |   20001 | 20.4 |   20001 | 19.7743403979155 |  0.625659602084532
 15 | {1,0.63796,0.00,8.140,0,0.5380,6.0960,84.50,4.4619,4,307.0,21.00,380.02,10.26}   |   20001 | 18.2 |   20001 | 18.7400800902121 |  0.540080090212125
 16 | {1,0.62739,0.00,8.140,0,0.5380,5.8340,56.50,4.4986,4,307.0,21.00,395.62,8.47}    |   20001 | 19.9 |   20001 | 19.6187933144569 |  0.281206685543061
 17 | {1,1.05393,0.00,8.140,0,0.5380,5.9350,29.30,4.4986,4,307.0,21.00,386.85,6.58}    |   20001 | 23.1 |   20001 | 23.3492239648177 |  0.249223964817737
 18 | {1,0.78420,0.00,8.140,0,0.5380,5.9900,81.70,4.2579,4,307.0,21.00,386.75,14.67}   |   20001 | 17.5 |   20001 | 17.0806608347814 |  0.419339165218577
 19 | {1,0.80271,0.00,8.140,0,0.5380,5.4560,36.60,3.7965,4,307.0,21.00,288.99,11.69}   |   20001 | 20.2 |   20001 | 20.1559086626409 |  0.044091337359113
 20 | {1,0.72580,0.00,8.140,0,0.5380,5.7270,69.50,3.7965,4,307.0,21.00,390.95,11.28}   |   20001 | 18.2 |   20001 | 18.6980897920022 |  0.498089792002183
(20 rows)
</pre>
RMS error:
<pre class="example">
SELECT SQRT(AVG((y-estimated_y)*(y-estimated_y))) as rms_error FROM lin_housing
JOIN mlp_regress_prediction USING (id);
</pre>
<pre class="result">
     rms_error
-------------------+
 0.912158035902468
(1 row)
</pre>

<h4>Regression with Grouping and Mini-Batching</h4>

-# To use grouping and mini-batching, we must first
re-run the preprocessor and specify grouping:
<pre class="example">
DROP TABLE IF EXISTS lin_housing_packed, lin_housing_packed_summary, lin_housing_packed_standardization;
SELECT madlib.minibatch_preprocessor('lin_housing',         -- Source table
                                     'lin_housing_packed',  -- Output table
                                     'y',                   -- Dependent variable
                                     'x',                   -- Independent variables
                                     'zipcode'              -- Group by zipcode
                                     );
</pre>

-# Train regression model and group the training data by zipcode
to learn a different model for each zipcode.
<pre class="example">
DROP TABLE IF EXISTS mlp_regress_group, mlp_regress_group_summary, mlp_regress_group_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
SELECT madlib.mlp_regression(
    'lin_housing_packed',    -- Source table
    'mlp_regress_group',     -- Desination table
    'independent_varname',   -- Input features
    'dependent_varname',     -- Dependent variable
    ARRAY[25,25],     -- Number of units per layer
    'learning_rate_init=0.001,
    n_iterations=500,
    lambda=0.001,
    tolerance=0',            -- Optimizer params
    'relu',                  -- Activation function
    NULL,                    -- Default weight (1)
    FALSE,                   -- No warm start
    FALSE,                   -- Not verbose
    'zipcode'                -- Grouping column
);
</pre>
View regression model with grouping:
<pre class="example">
\\x on
SELECT * FROM mlp_regress_group;
</pre>
<pre class="result">
-[ RECORD 1 ]--+------------------------------------------------------------------------------------
zipcode        | 200001
coeff          | {-0.193588485849,0.063428493184,-0.30440608833,-0.355695802004,-0.175942716164 ...
loss           | 0.0904009145541
num_iterations | 500
-[ RECORD 2 ]--+------------------------------------------------------------------------------------
zipcode        | 94016
coeff          | {-0.18965351506,0.0633650963628,-0.302423579808,-0.334367637252,-0.230043593847 ...
loss           | 1.04772100552
num_iterations | 500
</pre>

-# Prediction using the regression model for each group based on the zipcode:
<pre class="example">
DROP TABLE IF EXISTS mlp_regress_prediction;
SELECT madlib.mlp_predict(
         'mlp_regress_group',         -- Model table
         'lin_housing',             -- Test data table
         'id',                        -- Id column in test table
         'mlp_regress_prediction',    -- Output table for predictions
         'response'                   -- Output values, not probabilities
     );
\\x off
SELECT * FROM lin_housing JOIN mlp_regress_prediction USING (zipcode, id) ORDER BY zipcode, id;
</pre>
<pre class="result">
 zipcode | id |                                        x                                         |  y   |   estimated_y
---------+----+----------------------------------------------------------------------------------+------+------------------
   20001 | 12 | {1,0.11747,12.50,7.870,0,0.5240,6.0090,82.90,6.2267,5,311.0,15.20,396.90,13.27}  | 18.9 | 19.2272848285357
   20001 | 13 | {1,0.09378,12.50,7.870,0,0.5240,5.8890,39.00,5.4509,5,311.0,15.20,390.50,15.71}  | 21.7 | 21.3979318641202
   20001 | 14 | {1,0.62976,0.00,8.140,0,0.5380,5.9490,61.80,4.7075,4,307.0,21.00,396.90,8.26}    | 20.4 | 19.7743403979155
   20001 | 15 | {1,0.63796,0.00,8.140,0,0.5380,6.0960,84.50,4.4619,4,307.0,21.00,380.02,10.26}   | 18.2 | 18.7400800902121
   20001 | 16 | {1,0.62739,0.00,8.140,0,0.5380,5.8340,56.50,4.4986,4,307.0,21.00,395.62,8.47}    | 19.9 | 19.6187933144569
   20001 | 17 | {1,1.05393,0.00,8.140,0,0.5380,5.9350,29.30,4.4986,4,307.0,21.00,386.85,6.58}    | 23.1 | 23.3492239648177
   20001 | 18 | {1,0.78420,0.00,8.140,0,0.5380,5.9900,81.70,4.2579,4,307.0,21.00,386.75,14.67}   | 17.5 | 17.0806608347814
   20001 | 19 | {1,0.80271,0.00,8.140,0,0.5380,5.4560,36.60,3.7965,4,307.0,21.00,288.99,11.69}   | 20.2 | 20.1559086626409
   20001 | 20 | {1,0.72580,0.00,8.140,0,0.5380,5.7270,69.50,3.7965,4,307.0,21.00,390.95,11.28}   | 18.2 | 18.6980897920022
   94016 |  1 | {1,0.00632,18.00,2.310,0,0.5380,6.5750,65.20,4.0900,1,296.0,15.30,396.90,4.98}   |   24 | 23.9714991250013
   94016 |  2 | {1,0.02731,0.00,7.070,0,0.4690,6.4210,78.90,4.9671,2,242.0,17.80,396.90,9.14}    | 21.6 | 22.3655180133895
   94016 |  3 | {1,0.02729,0.00,7.070,0,0.4690,7.1850,61.10,4.9671,2,242.0,17.80,392.83,4.03}    | 34.7 | 33.8620767428645
   94016 |  4 | {1,0.03237,0.00,2.180,0,0.4580,6.9980,45.80,6.0622,3,222.0,18.70,394.63,2.94}    | 33.4 | 35.3094157686524
   94016 |  5 | {1,0.06905,0.00,2.180,0,0.4580,7.1470,54.20,6.0622,3,222.0,18.70,396.90,5.33}    | 36.2 | 35.0379122731818
   94016 |  6 | {1,0.02985,0.00,2.180,0,0.4580,6.4300,58.70,6.0622,3,222.0,18.70,394.12,5.21}    | 28.7 | 27.5207943492151
   94016 |  7 | {1,0.08829,12.50,7.870,0,0.5240,6.0120,66.60,5.5605,5,311.0,15.20,395.60,12.43}  | 22.9 | 24.9841422781166
   94016 |  8 | {1,0.14455,12.50,7.870,0,0.5240,6.1720,96.10,5.9505,5,311.0,15.20,396.90,19.15}  | 27.1 | 24.5403994064793
   94016 |  9 | {1,0.21124,12.50,7.870,0,0.5240,5.6310,100.00,6.0821,5,311.0,15.20,386.63,29.93} | 16.5 | 17.2588278443879
   94016 | 10 | {1,0.17004,12.50,7.870,0,0.5240,6.0040,85.90,6.5921,5,311.0,15.20,386.71,17.10}  | 18.9 | 17.0600407532569
   94016 | 11 | {1,0.22489,12.50,7.870,0,0.5240,6.3770,94.30,6.3467,5,311.0,15.20,392.52,20.45}  |   15 | 15.2284207930287
(20 rows)
</pre>
Note that the results you get for all examples may vary with the database you are using.

@anchor background
@par Technical Background

To train a neural net, the loss function is minimized using stochastic gradient descent.
In the case of classification, the loss function is cross entropy.  For regression, mean square error
is used. Weights in the neural net are updated via the backpropogation process, which uses dynamic
programming to compute the partial derivative of each weight with respect to the overall loss. This
partial derivative incorporates the activation function used, which requires that the
activation function be differentiable.

For an overview of multilayer perceptrons, see [1].

For details on backpropogation, see [2].

On the effect of database cluster size: as the database cluster
size increases, the per iteration loss will be higher since the
model only sees 1/n of the data, where n is the number of segments.
However, each iteration runs faster than single node because it is only
traversing 1/n of the data.  For large data sets, all else being equal,
a bigger cluster will achieve a given accuracy faster than a single node
although it may take more iterations to achieve that accuracy.

@anchor literature
@literature

@anchor mlp-lit-1
[1] https://en.wikipedia.org/wiki/Multilayer_perceptron

[2] Yu Hen Hu. "Lecture 11. MLP (III): Back-Propagation."
University of Wisconsin Madison: Computer-Aided Engineering. Web. 12 July 2017,
http://homepages.cae.wisc.edu/~ece539/videocourse/notes/pdf/lec%2011%20MLP%20(3)%20BP.pdf

[3] "Neural Networks for Machine Learning", Lectures 6a and 6b on mini-batch gradient descent,
Geoffrey Hinton with Nitish Srivastava and Kevin Swersky,
http://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf

@anchor related
@par Related Topics

File mlp.sql_in documenting the training function

*/

CREATE TYPE MADLIB_SCHEMA.mlp_result AS (
        coeff    DOUBLE PRECISION[],
        loss     DOUBLE PRECISION
);

--------------------------------------------------------------------------
-- create SQL functions for IGD optimizer
--------------------------------------------------------------------------
CREATE FUNCTION MADLIB_SCHEMA.mlp_igd_transition(
        state              DOUBLE PRECISION[],
        ind_var            DOUBLE PRECISION[],
        dep_var            DOUBLE PRECISION[],
        previous_state     DOUBLE PRECISION[],
        layer_sizes        DOUBLE PRECISION[],
        learning_rate_init DOUBLE PRECISION,
        activation         INTEGER,
        is_classification  INTEGER,
        weight             DOUBLE PRECISION,
        warm_start_coeff   DOUBLE PRECISION[],
        lambda             DOUBLE PRECISION,
        momentum           DOUBLE PRECISION,
        is_nesterov        BOOLEAN
    )
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE;

CREATE FUNCTION MADLIB_SCHEMA.mlp_minibatch_transition(
        state              DOUBLE PRECISION[],
        ind_var            DOUBLE PRECISION[],
        dep_var            DOUBLE PRECISION[],
        previous_state     DOUBLE PRECISION[],
        layer_sizes        DOUBLE PRECISION[],
        learning_rate_init DOUBLE PRECISION,
        activation         INTEGER,
        is_classification  INTEGER,
        weight             DOUBLE PRECISION,
        warm_start_coeff   DOUBLE PRECISION[],
        lambda             DOUBLE PRECISION,
        batch_size         INTEGER,
        n_epochs           INTEGER,
        momentum           DOUBLE PRECISION,
        is_nesterov        BOOLEAN
    )
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE;

CREATE FUNCTION MADLIB_SCHEMA.mlp_igd_merge(
        state1 DOUBLE PRECISION[],
        state2 DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT;

CREATE FUNCTION MADLIB_SCHEMA.mlp_igd_final(
        state DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT;

CREATE FUNCTION MADLIB_SCHEMA.mlp_minibatch_merge(
        state1 DOUBLE PRECISION[],
        state2 DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT;

CREATE FUNCTION MADLIB_SCHEMA.mlp_minibatch_final(
        state DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT;

/**
 * @internal
 * @brief Perform one iteration of backprop
 */
CREATE AGGREGATE MADLIB_SCHEMA.mlp_igd_step(
        /* ind_var */             DOUBLE PRECISION[],
        /* dep_var */             DOUBLE PRECISION[],
        /* previous_state */      DOUBLE PRECISION[],
        /* layer_sizes */         DOUBLE PRECISION[],
        /* learning_rate_init */  DOUBLE PRECISION,
        /* activation */          INTEGER,
        /* is_classification */   INTEGER,
        /* weight */              DOUBLE PRECISION,
        /* warm_start_coeff */    DOUBLE PRECISION[],
        /* lambda */              DOUBLE PRECISION,
        /* momentum */            DOUBLE PRECISION,
        /* is_nesterov */         BOOLEAN
        )(
    STYPE=DOUBLE PRECISION[],
    SFUNC=MADLIB_SCHEMA.mlp_igd_transition,
    m4_ifdef(`__POSTGRESQL__', `', `prefunc=MADLIB_SCHEMA.mlp_igd_merge,')
    FINALFUNC=MADLIB_SCHEMA.mlp_igd_final,
    INITCOND='{0,0,0,0,0,0,0,0,0,0,0,0}'
);
-------------------------------------------------------------------------

/**
 * @internal
 * @brief Perform one iteration of backprop
 */
CREATE AGGREGATE MADLIB_SCHEMA.mlp_minibatch_step(
        /* ind_var */             DOUBLE PRECISION[],
        /* dep_var */             DOUBLE PRECISION[],
        /* previous_state */      DOUBLE PRECISION[],
        /* layer_sizes */         DOUBLE PRECISION[],
        /* learning_rate_init */  DOUBLE PRECISION,
        /* activation */          INTEGER,
        /* is_classification */   INTEGER,
        /* weight */              DOUBLE PRECISION,
        /* warm_start_coeff */    DOUBLE PRECISION[],
        /* lambda */              DOUBLE PRECISION,
        /* batch_size */          INTEGER,
        /* n_epochs */            INTEGER,
        /* momentum */            DOUBLE PRECISION,
        /* is_nesterov */         BOOLEAN
        )(
    STYPE=DOUBLE PRECISION[],
    SFUNC=MADLIB_SCHEMA.mlp_minibatch_transition,
    m4_ifdef(`__POSTGRESQL__', `', `prefunc=MADLIB_SCHEMA.mlp_minibatch_merge,')
    FINALFUNC=MADLIB_SCHEMA.mlp_minibatch_final,
    INITCOND='{0,0,0,0,0,0,0,0,0,0,0,0,0,0}'
);
-------------------------------------------------------------------------

CREATE FUNCTION MADLIB_SCHEMA.internal_mlp_igd_distance(
    /*+ state1 */ DOUBLE PRECISION[],
    /*+ state2 */ DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE STRICT;

CREATE FUNCTION MADLIB_SCHEMA.internal_mlp_igd_result(
    /*+ state */ DOUBLE PRECISION[])
RETURNS MADLIB_SCHEMA.mlp_result AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE STRICT;

CREATE FUNCTION MADLIB_SCHEMA.internal_mlp_minibatch_result(
    /*+ state */ DOUBLE PRECISION[])
RETURNS MADLIB_SCHEMA.mlp_result AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE STRICT;
-------------------------------------------------------------------------

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR,
    weights              VARCHAR,
    warm_start           BOOLEAN,
    verbose              BOOLEAN,
    grouping_col         VARCHAR
) RETURNS VOID AS $$
    PythonFunctionBodyOnly(`convex', `mlp_igd')
    with AOControl(False):
        mlp_igd.mlp(schema_madlib,
                    source_table,
                    output_table,
                    independent_varname,
                    dependent_varname,
                    hidden_layer_sizes,
                    optimizer_params,
                    activation,
                    True,
                    weights,
                    warm_start,
                    verbose,
                    grouping_col)
$$ LANGUAGE plpythonu VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR,
    weights              VARCHAR,
    warm_start           BOOLEAN,
    verbose              BOOLEAN,
    grouping_col         VARCHAR
) RETURNS VOID AS $$
    PythonFunctionBodyOnly(`convex', `mlp_igd')
    with AOControl(False):
        mlp_igd.mlp(schema_madlib,
                    source_table,
                    output_table,
                    independent_varname,
                    dependent_varname,
                    hidden_layer_sizes,
                    optimizer_params,
                    activation,
                    False,
                    weights,
                    warm_start,
                    verbose,
                    grouping_col )
$$ LANGUAGE plpythonu VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR,
    weights              VARCHAR,
    warm_start           BOOLEAN,
    verbose              BOOLEAN
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_classification($1, $2, $3, $4, $5, $6, $7, $8, $9, $10, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR,
    weights              VARCHAR,
    warm_start           BOOLEAN
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_classification($1, $2, $3, $4, $5, $6, $7, $8, $9, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR,
    weights              VARCHAR
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_classification($1, $2, $3, $4, $5, $6, $7, $8, FALSE, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_classification($1, $2, $3, $4, $5, $6, $7, NULL, FALSE, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_classification($1, $2, $3, $4, $5, $6, NULL, NULL, FALSE, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[]
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_classification($1, $2, $3, $4, $5, NULL, NULL, NULL, FALSE, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR,
    weights              VARCHAR,
    warm_start           BOOLEAN,
    verbose              BOOLEAN
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_regression($1, $2, $3, $4, $5, $6, $7, $8, $9, $10, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR,
    weights              VARCHAR,
    warm_start           BOOLEAN
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_regression($1, $2, $3, $4, $5, $6, $7, $8, $9, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR,
    weights              VARCHAR
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_regression($1, $2, $3, $4, $5, $6, $7, $8, FALSE, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR,
    activation           VARCHAR
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_regression($1, $2, $3, $4, $5, $6, $7, NULL, FALSE, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[],
    optimizer_params     VARCHAR
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_regression($1, $2, $3, $4, $5, $6, NULL, NULL, FALSE, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression(
    source_table         VARCHAR,
    output_table         VARCHAR,
    independent_varname  VARCHAR,
    dependent_varname    VARCHAR,
    hidden_layer_sizes   INTEGER[]
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.mlp_regression($1, $2, $3, $4, $5, NULL, NULL, NULL, FALSE, FALSE, NULL);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_predict(
    model_table      VARCHAR,
    data_table      VARCHAR,
    id_col_name    VARCHAR,
    output_table      VARCHAR,
    pred_type      VARCHAR
) RETURNS VOID AS $$
    PythonFunctionBodyOnly(`convex', `mlp_igd')
    with AOControl(False):
        mlp_igd.mlp_predict(schema_madlib,
                            model_table,
                            data_table,
                            id_col_name,
                            output_table,
                            pred_type)
$$ LANGUAGE plpythonu VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');

CREATE FUNCTION MADLIB_SCHEMA.internal_predict_mlp(
        coeff DOUBLE PRECISION[],
        independent_varname DOUBLE PRECISION[],
        is_classification DOUBLE PRECISION,
        activation DOUBLE PRECISION,
        layer_sizes DOUBLE PRECISION[],
        is_response INTEGER,
        x_means DOUBLE PRECISION[],
        x_stds DOUBLE PRECISION[],
        array_dep_var_for_classification INTEGER
    )
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT;


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification(
   message TEXT
) RETURNS TEXT AS $$
    PythonFunctionBodyOnly(`convex', `mlp_igd')
    return mlp_igd.mlp_help(schema_madlib,message,True)
$$ LANGUAGE plpythonu
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_classification()
RETURNS TEXT AS $$
  SELECT MADLIB_SCHEMA.mlp_classification(NULL::TEXT)
$$ LANGUAGE SQL IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression(
   message TEXT
) RETURNS TEXT AS $$
    PythonFunctionBodyOnly(`convex', `mlp_igd')
    return mlp_igd.mlp_help(schema_madlib,message,False)
$$ LANGUAGE plpythonu
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_regression()
RETURNS TEXT AS $$
  SELECT MADLIB_SCHEMA.mlp_regression(NULL::TEXT)
$$ LANGUAGE SQL IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_predict(
   message TEXT
) RETURNS TEXT AS $$
    PythonFunctionBodyOnly(`convex', `mlp_igd')
    return mlp_igd.mlp_predict_help(schema_madlib,message)
$$ LANGUAGE plpythonu
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlp_predict()
RETURNS TEXT AS $$
  SELECT MADLIB_SCHEMA.mlp_predict(NULL::TEXT)
$$ LANGUAGE SQL IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');