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* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
/* eslint-disable no-restricted-syntax, no-invalid-this */
'use strict';
// MODULES //
var setReadOnly = require( '@stdlib/utils/define-nonenumerable-read-only-property' );
var setReadOnlyAccessor = require( '@stdlib/utils/define-nonenumerable-read-only-accessor' );
var format = require( '@stdlib/string/format' );
var gdot = require( '@stdlib/blas/base/gdot' ).ndarray;
var gaxpy = require( '@stdlib/blas/base/gaxpy' ).ndarray;
var dcopy = require( '@stdlib/blas/base/dcopy' );
var dscal = require( '@stdlib/blas/base/dscal' );
var max = require( '@stdlib/math/base/special/max' );
var exp = require( '@stdlib/math/base/special/exp' );
var pow = require( '@stdlib/math/base/special/pow' );
var sigmoid = require( '@stdlib/math/base/special/expit' );
var Float64Array = require( '@stdlib/array/float64' );
var ndarray = require( '@stdlib/ndarray/ctor' );
var shape2strides = require( '@stdlib/ndarray/base/shape2strides' );
var numel = require( '@stdlib/ndarray/base/numel' );
var vind2bind = require( '@stdlib/ndarray/base/vind2bind' );
// VARIABLES //
var MIN_SCALING_FACTOR = 1.0e-7;
var MIN_SCALE = 1.0e-11;
var LEARNING_RATE_METHODS = {
'basic': '_basicLearningRate',
'constant': '_constantLearningRate',
'invscaling': '_inverseScalingLearningRate',
'pegasos': '_pegasosLearningRate'
};
var LOSS_METHODS = {
'hinge': '_hingeLoss',
'log': '_logLoss',
'modified-huber': '_modifiedHuberLoss',
'perceptron': '_perceptronLoss',
'squared-hinge': '_squaredHingeLoss'
};
// MAIN //
/**
* Model constructor.
*
* ## Notes
*
* - The model (weight vector) implementation is inspired by the [sofia-ml][sofia-ml] library.
*
* [sofia-ml]: https://code.google.com/archive/p/sofia-ml/
*
* @private
* @constructor
* @param {PositiveInteger} N - number of feature weights (excluding bias/intercept term)
* @param {Options} opts - model options
* @param {PositiveNumber} opts.lambda - regularization parameter
* @param {ArrayLikeObject} opts.learningRate - learning rate function and associated parameters
* @param {string} opts.loss - loss function
* @param {boolean} opts.intercept - boolean indicating whether to include an intercept
* @returns {Model} model
*/
function Model( N, opts ) {
var len;
// Set internal properties:
this._N = N;
this._opts = opts;
this._scaleFactor = 1.0;
this._t = 0; // iteration counter (i.e., number of updates)
// Determine the learning rate function:
this._learningRateMethod = LEARNING_RATE_METHODS[ opts.learningRate[ 0 ] ];
// Determine the loss function:
this._lossMethod = LOSS_METHODS[ opts.loss ];
// Determine the number of model coefficients:
len = N;
if ( opts.intercept ) {
len += 1;
}
// Initialize a model weight vector with all weights set to zero:
this._weights = new Float64Array( len );
// Initialize model coefficients to zero:
this._coefficients = new ndarray( 'float64', new Float64Array( len ), [ len ], [ 1 ], 0, 'row-major' );
return this;
}
/**
* Adds a provided input vector to the model weight vector.
*
* @private
* @name _add
* @memberof Model.prototype
* @type {Function}
* @param {VectorLike} x - input vector
* @param {number} scale - scale factor
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, '_add', function add( x, scale ) {
var s = scale / this._scaleFactor;
var w = this._weights;
// Scale `x` and add to the model weight vector:
gaxpy( x.shape[ 0 ], s, x.data, x.strides[ 0 ], x.offset, w, 1, 0 );
// If an intercept is assumed, treat `x` as containing one additional element equal to one...
if ( this._opts.intercept ) {
w[ this._N ] += s;
}
return this;
});
/**
* Computes a learning rate.
*
* ## Notes
*
* - This learning rate function is based on the learning rate function of the same name in the [sofia-ml][sofia-ml] library.
*
* [sofia-ml]: https://code.google.com/archive/p/sofia-ml/
*
* @private
* @name _basicLearningRate
* @memberof Model.prototype
* @type {Function}
* @returns {number} learning rate
*/
setReadOnly( Model.prototype, '_basicLearningRate', function basic() {
return 10.0 / ( 10.0+this._t );
});
/**
* Returns a constant learning rate.
*
* @private
* @name _constantLearningRate
* @memberof Model.prototype
* @type {Function}
* @returns {number} learning rate
*/
setReadOnly( Model.prototype, '_constantLearningRate', function constant() {
return this._opts.learningRate[ 1 ];
});
/**
* Calculates the dot product of the model weight vector and a provided vector `x`.
*
* @private
* @name _dot
* @memberof Model.prototype
* @type {Function}
* @param {NumericArray} buf - ndarray data buffer
* @param {integer} stride - stride
* @param {NonNegativeInteger} offset - index offset
* @returns {number} dot product
*/
setReadOnly( Model.prototype, '_dot', function dot( buf, stride, offset ) {
var v = gdot( this._N, this._weights, 1, 0, buf, stride, offset );
if ( this._opts.intercept ) {
v += this._weights[ this._N ];
}
v *= this._scaleFactor;
return v;
});
/**
* Updates the model weight vector using the hinge loss function.
*
* ## Notes
*
* - The hinge loss function is defined as
*
* ```tex
* L(y, f(x)) = \max\{ 0, 1 - y\,f(x) \}
* ```
*
* where
*
* ```tex
* f(x) = w^T x + b
* ```
*
* with \\(w\\) being the model weight vector and \\(b\\) being the intercept.
*
* @private
* @name _hingeLoss
* @memberof Model.prototype
* @type {Function}
* @param {VectorLike} x - feature vector
* @param {integer} y - response value
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, '_hingeLoss', function hingeLoss( x, y ) {
var eta;
var d;
eta = this[ this._learningRateMethod ]();
this._regularize( eta );
d = this._dot( x.data, x.strides[ 0 ], x.offset );
if ( ( y*d ) < 1.0 ) {
this._add( x, y*eta );
}
return this;
});
/**
* Computes a learning rate according to an inverse scaling formula.
*
* ## Notes
*
* - The inverse scaling formula is defined as
*
* ```tex
* \eta = \frac{\eta_0}{t^{k}}
* ```
*
* where \\(\eta_0\\) is an initial learning rate, \\(t\\) is the current iteration, and \\(k\\) is an exponent controlling how quickly the learning rate decreases.
*
* @private
* @name _inverseScalingLearningRate
* @memberof Model.prototype
* @type {Function}
* @returns {number} learning rate
*/
setReadOnly( Model.prototype, '_inverseScalingLearningRate', function invscaling() {
var params = this._opts.learningRate;
return params[ 1 ] / pow( this._t, params[ 2 ] );
});
/**
* Updates the model weight vector using the log loss function.
*
* ## Notes
*
* - The log loss function is defined as
*
* ```tex
* L(y, f(x)) = \ln( 1 + \exp( -y\,f(x) ) )
* ```
*
* where
*
* ```tex
* f(x) = w^T x + b
* ```
*
* with \\(w\\) being the model weight vector and \\(b\\) being the intercept.
*
* @private
* @name _logLoss
* @memberof Model.prototype
* @type {Function}
* @param {VectorLike} x - feature vector
* @param {integer} y - response value
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, '_logLoss', function logLoss( x, y ) {
var loss;
var eta;
var d;
eta = this[ this._learningRateMethod ]();
this._regularize( eta );
d = this._dot( x.data, x.strides[ 0 ], x.offset );
loss = y / ( 1.0 + exp( y*d ) );
this._add( x, eta*loss );
return this;
});
/**
* Updates the model weight vector using the modified Huber loss function.
*
* ## Notes
*
* - The modified Huber loss function is defined as
*
* ```tex
* L(y, f(x)) = \begin{cases}
* \max(0, 1 - y\,f(x))^2 & \textrm{for}\,\,y\,f(x) \geq -1\\
* -4y\,f(x) & \textrm{otherwise}
* \end{cases}
* ```
*
* where
*
* ```tex
* f(x) = w^T x + b
* ```
*
* with \\(w\\) being the model weight vector and \\(b\\) being the intercept.
*
* ## References
*
* - Zhang, Tong. 2004. "Solving Large Scale Linear Prediction Problems Using Stochastic Gradient Descent Algorithms." In _Proceedings of the Twenty-First International Conference on Machine Learning_, 116. New York, NY, USA: Association for Computing Machinery. doi:[10.1145/1015330.1015332][@zhang:2004a].
*
* [@zhang:2004a]: https://doi.org/10.1145/1015330.1015332
*
* @private
* @name _modifiedHuberLoss
* @memberof Model.prototype
* @type {Function}
* @param {VectorLike} x - feature vector
* @param {integer} y - response value
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, '_modifiedHuberLoss', function modifiedHuber( x, y ) {
var eta;
var d;
eta = this[ this._learningRateMethod ]();
this._regularize( eta );
d = y * this._dot( x.data, x.strides[ 0 ], x.offset );
if ( d < -1.0 ) {
this._add( x, 4.0*eta*y );
} else {
this._add( x, eta*( y-(d*y) ) );
}
return this;
});
/**
* Computes a learning rate using Pegasos.
*
* ## References
*
* - Shalev-Shwartz, Shai, Yoram Singer, Nathan Srebro, and Andrew Cotter. 2011. "Pegasos: primal estimated sub-gradient solver for SVM." _Mathematical Programming_ 127 (1): 3–30. doi:[10.1007/s10107-010-0420-4][@shalevshwartz:2011a].
*
* [@shalevshwartz:2011a]: https://doi.org/10.1007/s10107-010-0420-4
*
* @private
* @name _pegasos
* @memberof Model.prototype
* @type {Function}
* @returns {number} learning rate
*/
setReadOnly( Model.prototype, '_pegasosLearningRate', function pegasos() {
return 1.0 / ( this._opts.lambda*this._t );
});
/**
* Updates the model weight vector using the perceptron loss function.
*
* ## Notes
*
* - The perceptron loss function is defined as
*
* ```tex
* L(y, f(x)) = \max(0, -y\,f(x))
* ```
*
* where
*
* ```tex
* f(x) = w^T x + b
* ```
*
* with \\(w\\) being the model weight vector and \\(b\\) being the intercept.
*
* - The perceptron loss function is equivalent to the hinge loss function without a margin.
*
* - The perceptron loss function does not update the model weight vector when the response is correctly classified.
*
* ## References
*
* - Rosenblatt, Frank. 1957. "The Perceptron–a perceiving and recognizing automaton." 85-460-1. Buffalo, NY, USA: Cornell Aeronautical Laboratory.
*
* @private
* @name _perceptronLoss
* @memberof Model.prototype
* @type {Function}
* @param {VectorLike} x - feature vector
* @param {integer} y - response value
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, '_perceptronLoss', function perceptron( x, y ) {
var eta;
var d;
eta = this[ this._learningRateMethod ]();
this._regularize( eta );
d = this._dot( x.data, x.strides[ 0 ], x.offset );
if ( ( y*d ) <= 0.0 ) {
this._add( x, y*eta );
}
return this;
});
/**
* Performs L2 regularization of the model weights.
*
* @private
* @name _regularize
* @memberof Model.prototype
* @type {Function}
* @param {PositiveNumber} eta - learning rate
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, '_regularize', function regularize( eta ) {
var lambda = this._opts.lambda;
if ( lambda <= 0.0 ) {
return this;
}
this._scale( max( 1.0-( eta*lambda ), MIN_SCALING_FACTOR ) );
return this;
});
/**
* Scale the model weight vector by a provided scaling factor.
*
* @private
* @name _scale
* @memberof Model.prototype
* @type {Function}
* @param {number} factor - scaling factor
* @throws {RangeError} scaling factor must be a positive number
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, '_scale', function scale( factor ) {
var s;
if ( factor <= 0.0 ) {
throw new RangeError( format( 'invalid argument. Attempting to scale a weight vector by a nonpositive value. This is likely due to too large a value of eta * lambda. Value: `%f`.', factor ) );
}
// Check whether we need to scale the weight vector to unity in order to avoid numerical issues...
s = this._scaleFactor;
if ( s < MIN_SCALE ) {
// Note: we only scale/shrink the feature weights, not the intercept...
dscal( this._N, s, this._weights, 1 );
this._scaleFactor = 1.0;
}
this._scaleFactor *= factor;
return this;
});
/**
* Updates the model weight vector using the squared hinge loss function.
*
* ## Notes
*
* - The squared hinge loss function is defined as
*
* ```tex
* L(y, f(x)) = \max\{ 0, 1 - y\,f(x) \}^2
* ```
*
* where
*
* ```tex
* f(x) = w^T x + b
* ```
*
* with \\(w\\) being the model weight vector and \\(b\\) being the intercept.
*
* @private
* @name _squaredHingeLoss
* @memberof Model.prototype
* @type {Function}
* @param {VectorLike} x - feature vector
* @param {integer} y - response value
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, '_squaredHingeLoss', function squaredHingeLoss( x, y ) {
var eta;
var d;
eta = this[ this._learningRateMethod ]();
this._regularize( eta );
d = y * this._dot( x.data, x.strides[ 0 ], x.offset );
if ( d < 1.0 ) {
this._add( x, eta*( y-(d*y) ) );
}
return this;
});
/**
* Returns the model coefficients.
*
* @private
* @name coefficients
* @memberof Model.prototype
* @type {Function}
* @returns {ndarray} model coefficients
*/
setReadOnlyAccessor( Model.prototype, 'coefficients', function coefficients() {
var c = this._coefficients.data;
var w = this._weights;
dcopy( w.length, w, 1, c, 1 );
dscal( this._N, this._scaleFactor, c, 1 );
return this._coefficients;
});
/**
* Returns the number of model features.
*
* @private
* @name nfeatures
* @memberof Model.prototype
* @type {PositiveInteger}
*/
setReadOnlyAccessor( Model.prototype, 'nfeatures', function nfeatures() {
return this._N;
});
/**
* Predicts the response value for one or more observation vectors `X`.
*
* @private
* @name predict
* @memberof Model.prototype
* @type {Function}
* @param {ndarray} X - feature vector
* @param {string} type - prediction type
* @returns {ndarray} ndarray containing response values
*/
setReadOnly( Model.prototype, 'predict', function predict( X, type ) {
var ndims;
var xbuf;
var ybuf;
var xsh;
var ysh;
var ord;
var ptr;
var sxn;
var sx;
var sy;
var ox;
var M;
var N;
var Y;
var v;
var i;
// Cache input array properties in case of lazy evaluation:
xbuf = X.data;
xsh = X.shape;
sx = X.strides;
ox = X.offset;
ord = X.order;
ndims = xsh.length - 1;
// The output array shape is the same as the input array shape without the last dimension (i.e., the number of dimensions is reduced by one)...
ysh = [];
for ( i = 0; i < ndims; i++ ) {
ysh.push( xsh[ i ] );
}
// Create an output array...
if ( ndims === 0 ) {
M = 1;
ybuf = new Float64Array( 1 );
sy = [ 0 ];
} else {
M = numel( ysh );
ybuf = new Float64Array( M );
sy = shape2strides( ysh, ord );
}
Y = new ndarray( 'int8', ybuf, ysh, sy, 0, ord );
// Loop over all observation vectors...
N = this._N; // number of features (i.e., size of last `X` dimension)
sxn = sx[ ndims ]; // stride of the last `X` dimension
for ( i = 0; i < M; i++ ) {
// Compute the index offset into the underlying data buffer pointing to the start of the current observation vector:
ptr = vind2bind( xsh, sx, ox, ord, i*N, 'throw' );
// Compute the dot product of the current observation vector with the model weight vector:
v = this._dot( xbuf, sxn, ptr );
// Determine the output value:
if ( type === 'label' ) {
v = ( v > 0 ) ? 1 : -1;
} else if ( type === 'probability' ) {
v = sigmoid( v );
} // else type === 'linear' (i.e., linear predictor)
// Set the element in the output array:
if ( ndims === 0 ) {
Y.iset( v );
} else {
Y.iset( i, v );
}
}
return Y;
});
/**
* Updates a model given a provided observation vector and response value.
*
* @private
* @name update
* @memberof Model.prototype
* @type {Function}
* @param {VectorLike} x - feature vector
* @param {integer} y - response value
* @returns {Model} model instance
*/
setReadOnly( Model.prototype, 'update', function update( x, y ) {
this._t += 1;
return this[ this._lossMethod ]( x, y );
});
// EXPORTS //
module.exports = Model;
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