1.18.0 User Documentation for Apache MADlib
Neural Network
Contents

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 Mini-Batch Preprocessor, 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.

Classification Training Function
The MLP classification training function has the following format:
mlp_classification(
source_table,
output_table,
independent_varname,
dependent_varname,
hidden_layer_sizes,
optimizer_params,
activation,
weights,
warm_start,
verbose,
grouping_col
)


Arguments

source_table

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.

output_table

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

independent_varname

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 Encoding Categorical Variables for more details.
dependent_varname

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.

hidden_layer_sizes (optional)

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.

optimizer_params (optional)

TEXT, default: NULL. Parameters for optimization in a comma-separated string of key-value pairs. See the description below for details.

activation (optional)

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.

weights (optional)
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.
warm_start (optional)

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.

verbose (optional)
BOOLEAN, default: FALSE. Provides verbose output of the results of training, including the value of loss at each iteration.
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.
grouping_col (optional)
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.

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

coeffs FLOAT8[]. Flat array containing the weights of the neural net. 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. FLOAT8. The cross entropy loss over the training data. See Technical Background section below for more details. If grouping_col is specified during training, a column for each grouping column is created.

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

source_table The source table. The independent variables. The dependent variable. The tolerance as given in optimizer_params. The initial learning rate as given in optimizer_params. The learning rate policy as given in optimizer_params. Momentum value as given in optimizer_params. Nesterov value as given in optimizer_params. The number of iterations run. The number of tries as given in optimizer_params. The number of units in each layer including the input and output layers. The activation function. True if the model was trained for classification, False if it was trained for regression. The classes which were trained against (empty for regression). The weight column used during training for giving different weights to different rows. NULL if no grouping_col was specified during training, and a comma-separated list of grouping column names if not.

A standardization table named <output_table>_standardization is also create, that has the following columns:

mean The mean for all input features (used for normalization). The standard deviation for all input features (used for normalization). If grouping_col is specified during training, a column for each grouping column is created.

Regression Training Function
The MLP regression training function has the following format:
mlp_regression(
source_table,
output_table,
independent_varname,
dependent_varname,
hidden_layer_sizes,
optimizer_params,
activation,
weights,
warm_start,
verbose,
grouping_col
)


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:

dependent_varname
TEXT. Name of the dependent variable column. For regression, supported types are any numeric type, or array of numeric types (for multiple regression).

Optimizer Parameters
Parameters in this section are supplied in the 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.
  'learning_rate_init = <value>,
learning_rate_policy = <value>,
gamma = <value>,
power = <value>,
iterations_per_step = <value>,
n_iterations = <value>,
n_tries = <value>,
lambda = <value>,
tolerance = <value>,
batch_size = <value>,
n_epochs = <value>,
momentum = <value>,
nesterov = <value>'


Optimizer Parameters

learning_rate_init

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.

learning_rate_policy

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))

gamma

Default: 0.1. Decay rate for learning rate when learning_rate_policy is 'exp' or 'step'.

power

Default: 0.5. Exponent for learning_rate_policy = 'inv'.

iterations_per_step

Default: 100. Number of iterations to run before decreasing the learning rate by a factor of gamma. Valid for learning rate policy = 'step'.

n_iterations

Default: 100. The maximum number of iterations allowed.

n_tries

Default: 1. Number of times to retrain the network with randomly initialized neural network weights.

lambda

Default: 0. The regularization coefficient for L2 regularization.

tolerance

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 tolerance or the iteration number is larger than n_iterations. If you want to run the full number of iterations specified in n_interations, set tolerance=0.0

batch_size

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.

n_epochs

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.

momentum

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.

nesterov

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.

Prediction Function
Used to generate predictions on novel data given a previously trained model. The same syntax is used for classification and regression.
mlp_predict(
model_table,
data_table,
id_col_name,
output_table,
pred_type
)


Arguments

model_table

TEXT. Model table produced by the training function.

data_table

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.

id_col_name

TEXT. The name of the id column in data_table.

output_table
TEXT. Name of the table where output predictions are written. If this table name is already in use, an error is returned. Table contains:
id Gives the 'id' for each prediction, corresponding to each row from the data_table. (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. (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.
pred_type
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.

Examples

Classification without Mini-Batching

1. Create an input data set.
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
(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');

2. 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.
DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
'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
);

View the model:
\x on
SELECT * FROM mlp_model;

-[ RECORD 1 ]--+------------------------------------------------------------------------------------
coeff          | {-0.40378996718,0.0157490328855,-0.298904053444,-0.984152185093,-0.657684089715 ...
loss           | 0.0103518565103
num_iterations | 500

View the model summary table:
SELECT * FROM mlp_model_summary;

-[ 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

View the model standardization table:
SELECT * FROM mlp_model_standardization;

-[ RECORD 1 ]------------------------------------------------------------------
mean | {5.45961538461539,2.99807692307692,3.025,0.851923076923077}
std  | {0.598799958694505,0.498262513685689,1.41840579525043,0.550346179381454}

3. 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.
DROP TABLE IF EXISTS mlp_prediction;
\x off
'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;

 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)

Count the misclassifications:
SELECT COUNT(*) FROM mlp_prediction JOIN iris_data USING (id)
WHERE mlp_prediction.estimated_class_text != iris_data.class_text;

 count
-------+
0


Classification with Mini-Batching

1. Use the same data set as above. Call mini-batch preprocessor:
DROP TABLE IF EXISTS iris_data_packed, iris_data_packed_summary, iris_data_packed_standardization;
'iris_data_packed',  -- Output table
'class_text',        -- Dependent variable
'attributes'        -- Independent variables
);

2. Train the classification model using similar parameters as before:
DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
'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
);

View the model:
\x on
SELECT * FROM mlp_model;

-[ RECORD 1 ]--+------------------------------------------------------------------------------------
coeff          | {-0.0780564661828377,-0.0781452670639994,0.3083605989842 ...
loss           | 0.00563534904146765
num_iterations | 500

3. 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.
DROP TABLE IF EXISTS mlp_prediction;
\x off
'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;

 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)

Count the misclassifications:
SELECT COUNT(*) FROM mlp_prediction JOIN iris_data USING (id)
WHERE mlp_prediction.estimated_class_text != iris_data.class_text;

 count
-------+
0


Classification with Other Parameters

1. 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.
DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
'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
);

View the model:
\x on
SELECT * FROM mlp_model;

-[ RECORD 1 ]--+------------------------------------------------------------------------------------
coeff          | {0.000156316559088915,0.131131017223563,-0.293990512682215 ...
loss           | 0.142238768280717
num_iterations | 50

2. 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.
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
);

View the model:
\x on
SELECT * FROM mlp_model;

-[ RECORD 1 ]--+------------------------------------------------------------------------------------
coeff          | {0.0883013960215441,0.235944854050211,-0.398126039487036 ...
loss           | 0.00818899646775659
num_iterations | 450

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.

Classification with Grouping

1. 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.
DROP TABLE IF EXISTS mlp_model_group, mlp_model_group_summary, mlp_model_group_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
'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
);

View the model:
\x on
SELECT * FROM mlp_model_group ORDER BY state;

-[ RECORD 1 ]--+------------------------------------------------------------------------------------
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

A separate model is learnt for each state, and the result table displays the name of the state (grouping column) associated with the model.
2. Prediction based on grouping using the state column:
DROP TABLE IF EXISTS mlp_prediction;
'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;

Result for the classification model:
   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)


Regression without Mini-Batching

1. Create a dataset with housing prices data.
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);

2. Now train a regression model using a multilayer perceptron with two hidden layers of twenty five nodes each:
DROP TABLE IF EXISTS mlp_regress, mlp_regress_summary, mlp_regress_standardization;
SELECT setseed(0);
'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
);

View the model:
\x on
SELECT * FROM mlp_regress;

[ RECORD 1 ]--+-------------------------------------------------------------------------------------
coeff          | {-0.250057620174,0.0630805938982,-0.290635490112,-0.382966162592,-0.212206338909...
loss           | 1.07042781236
num_iterations | 500

3. Prediction using the regression model:
DROP TABLE IF EXISTS mlp_regress_prediction;
'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
);

View results:
SELECT * FROM lin_housing JOIN mlp_regress_prediction USING (id) ORDER BY id;

 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)

RMS error:
SELECT SQRT(AVG((y-estimated_y)*(y-estimated_y))) as rms_error FROM lin_housing
JOIN mlp_regress_prediction USING (id);

    rms_error
------------------+
0.544960829104004


Regression with Mini-Batching

1. Call min-batch preprocessor using the same data set as above:
DROP TABLE IF EXISTS lin_housing_packed, lin_housing_packed_summary, lin_housing_packed_standardization;
'lin_housing_packed',  -- Output table
'y',                   -- Dependent variable
'x'                   -- Independent variables
);

2. Train regression model with mini-batching
DROP TABLE IF EXISTS mlp_regress, mlp_regress_summary, mlp_regress_standardization;
SELECT setseed(0);
'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
);

View model:
\x on
SELECT * FROM mlp_regress;

-[ RECORD 1 ]--+-------------------------------------------------------------
coeff          | {0.0395865908810001,-0.164860448878703,-0.132787863194324...
loss           | 0.0442383714892138
num_iterations | 500

3. Prediction for regression:
DROP TABLE IF EXISTS mlp_regress_prediction;
'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;

 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)

RMS error:
SELECT SQRT(AVG((y-estimated_y)*(y-estimated_y))) as rms_error FROM lin_housing
JOIN mlp_regress_prediction USING (id);

     rms_error
-------------------+
0.912158035902468
(1 row)


Regression with Grouping and Mini-Batching

1. To use grouping and mini-batching, we must first re-run the preprocessor and specify grouping:
DROP TABLE IF EXISTS lin_housing_packed, lin_housing_packed_summary, lin_housing_packed_standardization;
'lin_housing_packed',  -- Output table
'y',                   -- Dependent variable
'x',                   -- Independent variables
'zipcode'              -- Group by zipcode
);

2. Train regression model and group the training data by zipcode to learn a different model for each zipcode.
DROP TABLE IF EXISTS mlp_regress_group, mlp_regress_group_summary, mlp_regress_group_standardization;
-- Set seed so results are reproducible
SELECT setseed(0);
'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
);

View regression model with grouping:
\x on
SELECT * FROM mlp_regress_group;

-[ 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

3. Prediction using the regression model for each group based on the zipcode:
DROP TABLE IF EXISTS mlp_regress_prediction;
'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;

 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)

Note that the results you get for all examples may vary with the database you are using.

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.

Literature

[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

Related Topics

File mlp.sql_in documenting the training function