Neural Network

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

Also called "vanilla neural networks", 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.

- 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 )

**Arguments**

- source_table
TEXT. Name of the table containing the training data.

- 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.

- Note
- 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 on how to do this.

- 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.

- 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. Weights for input rows. Column name which specifies the weight for each input row. This weight will be incorporated into the update during stochastic gradient descent (SGD), but will 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.

- warm_start (optional)
BOOLEAN, default: FALSE. Initalize weights with the coefficients from the last call of the training function. If set to true, 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 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.

**Output tables**

The model table produced by MLP contains the following columns:

coeffs | FLOAT8[]. Flat array containing the weights of the neural net. |
---|---|

n_iterations | 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. |

loss | FLOAT8. The cross entropy over the training data. See Technical Background section below for more details. |

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

source_table | The source table. |
---|---|

independent_varname | The independent variables. |

dependent_varname | The dependent variable. |

tolerance | The tolerance as given in optimizer_params. |

learning_rate_init | The initial learning rate as given in optimizer_params. |

learning_rate_policy | The learning rate policy as given in optimizer_params. |

n_iterations | The number of iterations run. |

n_tries | The number of tries as given in optimizer_params. |

layer_sizes | The number of units in each layer including the input and output layers. |

activation | The activation function. |

is_classification | True if the model was trained for classification, False if it was trained for regression. |

classes | The classes which were trained against (empty for regression). |

weights | The weight column used during training. |

x_means | The mean for all input features (used for normalization). |

x_stds | The standard deviation for all input features (used for normalization). |

- 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 )

**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. 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>'

**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 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

- 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. estimated_COL_NAME (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. prob_CLASS (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
- Create an input data set.
DROP TABLE IF EXISTS iris_data; CREATE TABLE iris_data( id integer, attributes numeric[], class_text varchar, class integer ); INSERT INTO iris_data VALUES (1,ARRAY[5.1,3.5,1.4,0.2],'Iris-setosa',1), (2,ARRAY[4.9,3.0,1.4,0.2],'Iris-setosa',1), (3,ARRAY[4.7,3.2,1.3,0.2],'Iris-setosa',1), (4,ARRAY[4.6,3.1,1.5,0.2],'Iris-setosa',1), (5,ARRAY[5.0,3.6,1.4,0.2],'Iris-setosa',1), (6,ARRAY[5.4,3.9,1.7,0.4],'Iris-setosa',1), (7,ARRAY[4.6,3.4,1.4,0.3],'Iris-setosa',1), (8,ARRAY[5.0,3.4,1.5,0.2],'Iris-setosa',1), (9,ARRAY[4.4,2.9,1.4,0.2],'Iris-setosa',1), (10,ARRAY[4.9,3.1,1.5,0.1],'Iris-setosa',1), (11,ARRAY[7.0,3.2,4.7,1.4],'Iris-versicolor',2), (12,ARRAY[6.4,3.2,4.5,1.5],'Iris-versicolor',2), (13,ARRAY[6.9,3.1,4.9,1.5],'Iris-versicolor',2), (14,ARRAY[5.5,2.3,4.0,1.3],'Iris-versicolor',2), (15,ARRAY[6.5,2.8,4.6,1.5],'Iris-versicolor',2), (16,ARRAY[5.7,2.8,4.5,1.3],'Iris-versicolor',2), (17,ARRAY[6.3,3.3,4.7,1.6],'Iris-versicolor',2), (18,ARRAY[4.9,2.4,3.3,1.0],'Iris-versicolor',2), (19,ARRAY[6.6,2.9,4.6,1.3],'Iris-versicolor',2), (20,ARRAY[5.2,2.7,3.9,1.4],'Iris-versicolor',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; -- 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 TRUE -- Verbose );

- View the classification model.
-- Set extended display on for easier reading of output \x ON -- Results may vary depending on platform SELECT * FROM mlp_model;

Result:[ RECORD 1 ]--+--------------------------------------------------------------------------------------- coeff | {-0.172392477419,-0.0836446652758,-0.0162194484142,-0.647268294231,-0.504884325538... loss | 0.0136695756314 num_iterations | 500

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

- Now train a regression model using a multilayer perceptron with 2 hidden layers of 25 nodes each.
DROP TABLE IF EXISTS mlp_regress, mlp_regress_summary; 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', 'relu', NULL, -- Default weight (1) FALSE, -- No warm start TRUE -- Verbose );

- View the regression model.
-- Set extended display on for easier reading of output. \x ON -- Results may vary depending on platform. SELECT * FROM mlp_regress;

Result:[ RECORD 1 ]--+----------------------------------------------------------------------------------- coeff | {-0.135647108464,0.0315402969485,-0.117580589352,-0.23084537701,-0.10868726702... loss | 0.114125125042 num_iterations | 500

- Now let's look at the prediction functions. In the following examples we will use the training data set for prediction as well, which is not usual but serves to show the syntax. First we will test the classification example. The prediction is in the the estimated_class_text column with the actual value in the class_text column.
DROP TABLE IF EXISTS mlp_prediction; 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;

Result for the classification model:id | estimated_class_text | attributes | class_text | class ----+----------------------+-------------------+-----------------+------- 1 | Iris-setosa | {5.1,3.5,1.4,0.2} | Iris-setosa | 1 2 | Iris-setosa | {4.9,3.0,1.4,0.2} | Iris-setosa | 1 3 | Iris-setosa | {4.7,3.2,1.3,0.2} | Iris-setosa | 1 4 | Iris-setosa | {4.6,3.1,1.5,0.2} | Iris-setosa | 1 5 | Iris-setosa | {5.0,3.6,1.4,0.2} | Iris-setosa | 1 6 | Iris-setosa | {5.4,3.9,1.7,0.4} | Iris-setosa | 1 7 | Iris-setosa | {4.6,3.4,1.4,0.3} | Iris-setosa | 1 8 | Iris-setosa | {5.0,3.4,1.5,0.2} | Iris-setosa | 1 9 | Iris-setosa | {4.4,2.9,1.4,0.2} | Iris-setosa | 1 10 | Iris-setosa | {4.9,3.1,1.5,0.1} | Iris-setosa | 1 11 | Iris-versicolor | {7.0,3.2,4.7,1.4} | Iris-versicolor | 2 12 | Iris-versicolor | {6.4,3.2,4.5,1.5} | Iris-versicolor | 2 13 | Iris-versicolor | {6.9,3.1,4.9,1.5} | Iris-versicolor | 2 14 | Iris-versicolor | {5.5,2.3,4.0,1.3} | Iris-versicolor | 2 15 | Iris-versicolor | {6.5,2.8,4.6,1.5} | Iris-versicolor | 2 16 | Iris-versicolor | {5.7,2.8,4.5,1.3} | Iris-versicolor | 2 17 | Iris-versicolor | {6.3,3.3,4.7,1.6} | Iris-versicolor | 2 18 | Iris-versicolor | {4.9,2.4,3.3,1.0} | Iris-versicolor | 2 19 | Iris-versicolor | {6.6,2.9,4.6,1.3} | Iris-versicolor | 2 20 | Iris-versicolor | {5.2,2.7,3.9,1.4} | Iris-versicolor | 2

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

count -------+ 0

- Prediction using the regression model:
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 ); SELECT *, ABS(y-estimated_y) as abs_diff FROM lin_housing JOIN mlp_regress_prediction USING (id) ORDER BY id;

Result for the regression model:id | x | grp_by_col | y | estimated_y | abs_diff ----+-----------------------------------------------------------------------+------------+------+------------------+--------------------- 1 | {0.00632,18,2.31,0,0.538,6.575,65.2,4.09,1,296,15.3,396.9,4.98} | 1 | 24 | 23.9976935779896 | 0.00230642201042741 2 | {0.02731,0,7.07,0,0.469,6.421,78.9,4.9671,2,242,17.8,396.9,9.14} | 1 | 21.6 | 22.0225551503712 | 0.422555150371196 3 | {0.02729,0,7.07,0,0.469,7.185,61.1,4.9671,2,242,17.8,392.83,4.03} | 1 | 34.7 | 34.3269436787012 | 0.373056321298805 4 | {0.03237,0,2.18,0,0.458,6.998,45.8,6.0622,3,222,18.7,394.63,2.94} | 1 | 33.4 | 34.7421700032985 | 1.34217000329847 5 | {0.06905,0,2.18,0,0.458,7.147,54.2,6.0622,3,222,18.7,396.9,5.33} | 1 | 36.2 | 35.1914922401243 | 1.00850775987566 6 | {0.02985,0,2.18,0,0.458,6.43,58.7,6.0622,3,222,18.7,394.12,5.21} | 1 | 28.7 | 29.5286073543722 | 0.828607354372203 7 | {0.08829,12.5,7.87,0,0.524,6.012,66.6,5.5605,5,311,15.2,395.6,12.43} | 1 | 22.9 | 23.2022360304219 | 0.302236030421945 8 | {0.14455,12.5,7.87,0,0.524,6.172,96.1,5.9505,5,311,15.2,396.9,19.15} | 1 | 27.1 | 23.3649065290002 | 3.73509347099978 9 | {0.21124,12.5,7.87,0,0.524,5.631,100,6.0821,5,311,15.2,386.63,29.93} | 1 | 16.5 | 17.7779926866502 | 1.27799268665021 10 | {0.17004,12.5,7.87,0,0.524,6.004,85.9,6.5921,5,311,15.2,386.71,17.1} | 1 | 18.9 | 13.9266690257803 | 4.97333097421974 11 | {0.22489,12.5,7.87,0,0.524,6.377,94.3,6.3467,5,311,15.2,392.52,20.45} | 1 | 15 | 18.5049155838719 | 3.50491558387192 12 | {0.11747,12.5,7.87,0,0.524,6.009,82.9,6.2267,5,311,15.2,396.9,13.27} | 1 | 18.9 | 18.4287114359317 | 0.47128856406826 13 | {0.09378,12.5,7.87,0,0.524,5.889,39,5.4509,5,311,15.2,390.5,15.71} | 1 | 21.7 | 22.6228336114696 | 0.922833611469631 14 | {0.62976,0,8.14,0,0.538,5.949,61.8,4.7075,4,307,21,396.9,8.26} | 1 | 20.4 | 20.1083536059151 | 0.291646394084896 15 | {0.63796,0,8.14,0,0.538,6.096,84.5,4.4619,4,307,21,380.02,10.26} | 1 | 18.2 | 18.8935467873061 | 0.693546787306062 16 | {0.62739,0,8.14,0,0.538,5.834,56.5,4.4986,4,307,21,395.62,8.47} | 1 | 19.9 | 19.8383202293121 | 0.0616797706878742 17 | {1.05393,0,8.14,0,0.538,5.935,29.3,4.4986,4,307,21,386.85,6.58} | 1 | 23.1 | 23.160463540176 | 0.0604635401760412 18 | {0.7842,0,8.14,0,0.538,5.99,81.7,4.2579,4,307,21,386.75,14.67} | 1 | 17.5 | 16.8540384345856 | 0.64596156541436 19 | {0.80271,0,8.14,0,0.538,5.456,36.6,3.7965,4,307,21,288.99,11.69} | 1 | 20.2 | 20.3628760580577 | 0.162876058057684 20 | {0.7258,0,8.14,0,0.538,5.727,69.5,3.7965,4,307,21,390.95,11.28} | 1 | 18.2 | 18.1198369917265 | 0.0801630082734555 (20 rows)

RMS error:SELECT SQRT(SUM(ABS(y-estimated_y))/COUNT(y)) as rms_error FROM lin_housing JOIN mlp_regress_prediction USING (id);

rms_error ------------------+ 1.02862119016012

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

- Create an input data set.

- 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].

- Literature

[1] "Multilayer Perceptron." Wikipedia. Wikimedia Foundation, 12 July 2017. Web. 12 July 2017.

[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

- Related Topics

File mlp.sql_in documenting the training function