Stochastic Gradient Descent is a technique for minimizing a function which can be expressed as a sum of other functions. More...

Public Member Functions
	SGD (DecomposableFunctionType &function, const double stepSize=0.01, const size_t maxIterations=100000, const double tolerance=1e-5, const bool shuffle=true)
	Construct the SGD optimizer with the given function and parameters. More...

const DecomposableFunctionType &	Function () const
	Get the instantiated function to be optimized. More...

DecomposableFunctionType &	Function ()
	Modify the instantiated function. More...

size_t	MaxIterations () const
	Get the maximum number of iterations (0 indicates no limit). More...

size_t &	MaxIterations ()
	Modify the maximum number of iterations (0 indicates no limit). More...

double	Optimize (arma::mat &iterate)
	Optimize the given function using stochastic gradient descent. More...

template<>
double	Optimize (arma::mat &parameters)
	Template specialization for SGD optimizer. More...

bool	Shuffle () const
	Get whether or not the individual functions are shuffled. More...

bool &	Shuffle ()
	Modify whether or not the individual functions are shuffled. More...

double	StepSize () const
	Get the step size. More...

double &	StepSize ()
	Modify the step size. More...

double	Tolerance () const
	Get the tolerance for termination. More...

double &	Tolerance ()
	Modify the tolerance for termination. More...

Private Attributes
DecomposableFunctionType &	function
	The instantiated function. More...

size_t	maxIterations
	The maximum number of allowed iterations. More...

bool	shuffle
	Controls whether or not the individual functions are shuffled when iterating. More...

double	stepSize
	The step size for each example. More...

double	tolerance
	The tolerance for termination. More...

Detailed Description

template<typename DecomposableFunctionType>
class mlpack::optimization::SGD< DecomposableFunctionType >

Stochastic Gradient Descent is a technique for minimizing a function which can be expressed as a sum of other functions.

That is, suppose we have

$f(A) = \sum_{i = 0}^{n} f_i(A)$

and our task is to minimize $ A $ . Stochastic gradient descent iterates over each function $ f_i(A) $ , producing the following update scheme:

$A_{j + 1} = A_j + \alpha \nabla f_i(A)$

where $\alpha$ is a parameter which specifies the step size. $ i $ is chosen according to $ j $ (the iteration number). The SGD class supports either scanning through each of the $ n $ functions $ f_i(A) $ linearly, or in a random sequence. The algorithm continues until $ j $ reaches the maximum number of iterations—or when a full sequence of updates through each of the $ n $ functions produces an improvement within a certain tolerance $\epsilon$ . That is,

$| f(A_{j + n}) - f(A_j) | < \epsilon.$

The parameter $\epsilon$ is specified by the tolerance parameter to the constructor; $n$ is specified by the maxIterations parameter.

This class is useful for data-dependent functions whose objective function can be expressed as a sum of objective functions operating on an individual point. Then, SGD considers the gradient of the objective function operating on an individual point in its update of $ A $ .

For SGD to work, a DecomposableFunctionType template parameter is required. This class must implement the following function:

size_t NumFunctions(); double Evaluate(const arma::mat& coordinates, const size_t i); void Gradient(const arma::mat& coordinates, const size_t i, arma::mat& gradient);

NumFunctions() should return the number of functions ( $n$ ), and in the other two functions, the parameter i refers to which individual function (or gradient) is being evaluated. So, for the case of a data-dependent function, such as NCA (see mlpack::nca::NCA), NumFunctions() should return the number of points in the dataset, and Evaluate(coordinates, 0) will evaluate the objective function on the first point in the dataset (presumably, the dataset is held internally in the DecomposableFunctionType).

Template Parameters

DecomposableFunctionType Decomposable objective function type to be minimized.

Definition at line 76 of file sgd.hpp.

Constructor & Destructor Documentation

template<typename DecomposableFunctionType >

mlpack::optimization::SGD< DecomposableFunctionType >::SGD	(	DecomposableFunctionType &	function,
		const double	stepSize = `0.01`,
		const size_t	maxIterations = `100000`,
		const double	tolerance = `1e-5`,
		const bool	shuffle = `true`
	)

Construct the SGD optimizer with the given function and parameters.

The defaults here are not necessarily good for the given problem, so it is suggested that the values used be tailored to the task at hand. The maximum number of iterations refers to the maximum number of points that are processed (i.e., one iteration equals one point; one iteration does not equal one pass over the dataset).

Parameters

function	Function to be optimized (minimized).
stepSize	Step size for each iteration.
maxIterations	Maximum number of iterations allowed (0 means no limit).
tolerance	Maximum absolute tolerance to terminate algorithm.
shuffle	If true, the function order is shuffled; otherwise, each function is visited in linear order.

Member Function Documentation

template<typename DecomposableFunctionType >

const DecomposableFunctionType& mlpack::optimization::SGD< DecomposableFunctionType >::Function ( ) const

inline

Get the instantiated function to be optimized.

Definition at line 112 of file sgd.hpp.

template<typename DecomposableFunctionType >

DecomposableFunctionType& mlpack::optimization::SGD< DecomposableFunctionType >::Function ( )

inline

Modify the instantiated function.

Definition at line 114 of file sgd.hpp.

template<typename DecomposableFunctionType >

size_t mlpack::optimization::SGD< DecomposableFunctionType >::MaxIterations ( ) const

inline

Get the maximum number of iterations (0 indicates no limit).

Definition at line 122 of file sgd.hpp.

References mlpack::optimization::SGD< DecomposableFunctionType >::maxIterations.

template<typename DecomposableFunctionType >

size_t& mlpack::optimization::SGD< DecomposableFunctionType >::MaxIterations ( )

inline

Modify the maximum number of iterations (0 indicates no limit).

Definition at line 124 of file sgd.hpp.

References mlpack::optimization::SGD< DecomposableFunctionType >::maxIterations.

template<typename DecomposableFunctionType >

double mlpack::optimization::SGD< DecomposableFunctionType >::Optimize ( arma::mat & iterate )

Optimize the given function using stochastic gradient descent.

The given starting point will be modified to store the finishing point of the algorithm, and the final objective value is returned.

Parameters

iterate Starting point (will be modified).

Returns: Objective value of the final point.

template<>

double mlpack::optimization::SGD< mlpack::svd::RegularizedSVDFunction >::Optimize ( arma::mat & parameters )

Template specialization for SGD optimizer.

Used because the gradient affects only a small number of parameters per example, and thus the normal abstraction does not work as fast as we might like it to.

template<typename DecomposableFunctionType >

bool mlpack::optimization::SGD< DecomposableFunctionType >::Shuffle ( ) const

inline

Get whether or not the individual functions are shuffled.

Definition at line 132 of file sgd.hpp.

References mlpack::optimization::SGD< DecomposableFunctionType >::shuffle.

template<typename DecomposableFunctionType >

bool& mlpack::optimization::SGD< DecomposableFunctionType >::Shuffle ( )

inline

Modify whether or not the individual functions are shuffled.

Definition at line 134 of file sgd.hpp.

References mlpack::optimization::SGD< DecomposableFunctionType >::shuffle.

template<typename DecomposableFunctionType >

double mlpack::optimization::SGD< DecomposableFunctionType >::StepSize ( ) const

inline

Get the step size.

Definition at line 117 of file sgd.hpp.

References mlpack::optimization::SGD< DecomposableFunctionType >::stepSize.

template<typename DecomposableFunctionType >

double& mlpack::optimization::SGD< DecomposableFunctionType >::StepSize ( )

inline

Modify the step size.

Definition at line 119 of file sgd.hpp.

References mlpack::optimization::SGD< DecomposableFunctionType >::stepSize.

template<typename DecomposableFunctionType >

double mlpack::optimization::SGD< DecomposableFunctionType >::Tolerance ( ) const

inline

Get the tolerance for termination.

Definition at line 127 of file sgd.hpp.

References mlpack::optimization::SGD< DecomposableFunctionType >::tolerance.

template<typename DecomposableFunctionType >

double& mlpack::optimization::SGD< DecomposableFunctionType >::Tolerance ( )

inline

Modify the tolerance for termination.

Definition at line 129 of file sgd.hpp.

References mlpack::optimization::SGD< DecomposableFunctionType >::tolerance.

Member Data Documentation

template<typename DecomposableFunctionType >

DecomposableFunctionType& mlpack::optimization::SGD< DecomposableFunctionType >::function

private

The instantiated function.

Definition at line 138 of file sgd.hpp.

template<typename DecomposableFunctionType >

size_t mlpack::optimization::SGD< DecomposableFunctionType >::maxIterations

private

The maximum number of allowed iterations.

Definition at line 144 of file sgd.hpp.

Referenced by mlpack::optimization::SGD< DecomposableFunctionType >::MaxIterations().

template<typename DecomposableFunctionType >

bool mlpack::optimization::SGD< DecomposableFunctionType >::shuffle

private

Controls whether or not the individual functions are shuffled when iterating.

Definition at line 151 of file sgd.hpp.

Referenced by mlpack::optimization::SGD< DecomposableFunctionType >::Shuffle().

template<typename DecomposableFunctionType >

double mlpack::optimization::SGD< DecomposableFunctionType >::stepSize

private

The step size for each example.

Definition at line 141 of file sgd.hpp.

Referenced by mlpack::optimization::SGD< DecomposableFunctionType >::StepSize().

template<typename DecomposableFunctionType >

double mlpack::optimization::SGD< DecomposableFunctionType >::tolerance

private

The tolerance for termination.

Definition at line 147 of file sgd.hpp.

Referenced by mlpack::optimization::SGD< DecomposableFunctionType >::Tolerance().

The documentation for this class was generated from the following file:

src/mlpack/core/optimizers/sgd/sgd.hpp

Public Member Functions

Private Attributes

Detailed Description

template<typename DecomposableFunctionType> class mlpack::optimization::SGD< DecomposableFunctionType >

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename DecomposableFunctionType>
class mlpack::optimization::SGD< DecomposableFunctionType >