Stochastic gradient descent is a technique for minimizing a function which can be expressed as a sum of other 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. const DecomposableFunctionType & Function () const
Get the instantiated function to be optimized. DecomposableFunctionType & Function ()
Modify the instantiated function. size_t MaxIterations () const
Get the maximum number of iterations (0 indicates no limit). size_t & MaxIterations ()
Modify the maximum number of iterations (0 indicates no limit). double Optimize (arma::mat &iterate)
Optimize the given function using stochastic gradient descent. template<> double Optimize (arma::mat ¶meters)
bool Shuffle () const
Get whether or not the individual functions are shuffled. bool & Shuffle ()
Modify whether or not the individual functions are shuffled. double StepSize () const
Get the step size. double & StepSize ()
Modify the step size. double Tolerance () const
Get the tolerance for termination. double & Tolerance ()
Modify the tolerance for termination. std::string ToString () const
DecomposableFunctionType & function
The instantiated function. size_t maxIterations
The maximum number of allowed iterations. bool shuffle
Controls whether or not the individual functions are shuffled when iterating. double stepSize
The step size for each example. double tolerance
The tolerance for termination.
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
\[ f(A) = \sum_{i = 0}^{n} f_i(A) \].PP and our task is to minimize $ A $. Stochastic gradient descent iterates over each function $ f_i(A) $, producing the following update scheme:
\[ A_{j + 1} = A_j + \alpha \nabla f_i(A) \].PP 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 $ f_i(A) $ produces an improvement within a certain tolerance $ \psilon $. That is,
\[ | f(A_{j + n}) - f(A_j) | < \psilon. \].PP The parameter $\psilon$ 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 86 of file sgd.hpp.
Construct the SGD optimizer with the given function and parameters.
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.
Get the instantiated function to be optimized.
Definition at line 117 of file sgd.hpp.
Modify the instantiated function.
Definition at line 119 of file sgd.hpp.
Get the maximum number of iterations (0 indicates no limit).
Definition at line 127 of file sgd.hpp.
Modify the maximum number of iterations (0 indicates no limit).
Definition at line 129 of file sgd.hpp.
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.
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.
Get whether or not the individual functions are shuffled.
Definition at line 137 of file sgd.hpp.
Modify whether or not the individual functions are shuffled.
Definition at line 139 of file sgd.hpp.
Get the step size.
Definition at line 122 of file sgd.hpp.
Modify the step size.
Definition at line 124 of file sgd.hpp.
Get the tolerance for termination.
Definition at line 132 of file sgd.hpp.
Modify the tolerance for termination.
Definition at line 134 of file sgd.hpp.
The instantiated function.
Definition at line 146 of file sgd.hpp.
The maximum number of allowed iterations.
Definition at line 152 of file sgd.hpp.
Referenced by mlpack::optimization::SGD< mlpack::svd::RegularizedSVDFunction >::MaxIterations().
Controls whether or not the individual functions are shuffled when iterating.
Definition at line 159 of file sgd.hpp.
Referenced by mlpack::optimization::SGD< mlpack::svd::RegularizedSVDFunction >::Shuffle().
The step size for each example.
Definition at line 149 of file sgd.hpp.
Referenced by mlpack::optimization::SGD< mlpack::svd::RegularizedSVDFunction >::StepSize().
The tolerance for termination.
Definition at line 155 of file sgd.hpp.
Referenced by mlpack::optimization::SGD< mlpack::svd::RegularizedSVDFunction >::Tolerance().
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