internal/ceres/line_search.h - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2015 Google Inc. All rights reserved.
 // http://ceres-solver.org/
 //
 // Redistribution and use in source and binary forms, with or without
 // modification, are permitted provided that the following conditions are met:
 //
 // * Redistributions of source code must retain the above copyright notice,
 //   this list of conditions and the following disclaimer.
 // * Redistributions in binary form must reproduce the above copyright notice,
 //   this list of conditions and the following disclaimer in the documentation
 //   and/or other materials provided with the distribution.
 // * Neither the name of Google Inc. nor the names of its contributors may be
 //   used to endorse or promote products derived from this software without
 //   specific prior written permission.
 //
 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
 // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 // POSSIBILITY OF SUCH DAMAGE.
 //
 // Author: sameeragarwal@google.com (Sameer Agarwal)
 //
 // Interface for and implementation of various Line search algorithms.

 #ifndef CERES_INTERNAL_LINE_SEARCH_H_
 #define CERES_INTERNAL_LINE_SEARCH_H_

 #include <string>
 #include <vector>
 #include "ceres/function_sample.h"
 #include "ceres/internal/eigen.h"
 #include "ceres/internal/port.h"
 #include "ceres/types.h"

 namespace ceres {
 namespace internal {

 class Evaluator;
 class LineSearchFunction;

 // Line search is another name for a one dimensional optimization
 // algorithm. The name "line search" comes from the fact one
 // dimensional optimization problems that arise as subproblems of
 // general multidimensional optimization problems.
 //
 // While finding the exact minimum of a one dimensional function is
 // hard, instances of LineSearch find a point that satisfies a
 // sufficient decrease condition. Depending on the particular
 // condition used, we get a variety of different line search
 // algorithms, e.g., Armijo, Wolfe etc.
 class LineSearch {
  public:
   struct Summary;

   struct Options {
     // Degree of the polynomial used to approximate the objective
     // function.
     LineSearchInterpolationType interpolation_type = CUBIC;

     // Armijo and Wolfe line search parameters.

     // Solving the line search problem exactly is computationally
     // prohibitive. Fortunately, line search based optimization
     // algorithms can still guarantee convergence if instead of an
     // exact solution, the line search algorithm returns a solution
     // which decreases the value of the objective function
     // sufficiently. More precisely, we are looking for a step_size
     // s.t.
     //
     //  f(step_size) <= f(0) + sufficient_decrease * f'(0) * step_size
     double sufficient_decrease = 1e-4;

     // In each iteration of the Armijo / Wolfe line search,
     //
     // new_step_size >= max_step_contraction * step_size
     //
     // Note that by definition, for contraction:
     //
     //  0 < max_step_contraction < min_step_contraction < 1
     //
     double max_step_contraction = 1e-3;

     // In each iteration of the Armijo / Wolfe line search,
     //
     // new_step_size <= min_step_contraction * step_size
     // Note that by definition, for contraction:
     //
     //  0 < max_step_contraction < min_step_contraction < 1
     //
     double min_step_contraction = 0.9;

     // If during the line search, the step_size falls below this
     // value, it is truncated to zero.
     double min_step_size = 1e-9;

     // Maximum number of trial step size iterations during each line search,
     // if a step size satisfying the search conditions cannot be found within
     // this number of trials, the line search will terminate.
     int max_num_iterations = 20;

     // Wolfe-specific line search parameters.

     // The strong Wolfe conditions consist of the Armijo sufficient
     // decrease condition, and an additional requirement that the
     // step-size be chosen s.t. the _magnitude_ ('strong' Wolfe
     // conditions) of the gradient along the search direction
     // decreases sufficiently. Precisely, this second condition
     // is that we seek a step_size s.t.
     //
     //   |f'(step_size)| <= sufficient_curvature_decrease * |f'(0)|
     //
     // Where f() is the line search objective and f'() is the derivative
     // of f w.r.t step_size (d f / d step_size).
     double sufficient_curvature_decrease = 0.9;

     // During the bracketing phase of the Wolfe search, the step size is
     // increased until either a point satisfying the Wolfe conditions is
     // found, or an upper bound for a bracket containing a point satisfying
     // the conditions is found.  Precisely, at each iteration of the
     // expansion:
     //
     //   new_step_size <= max_step_expansion * step_size.
     //
     // By definition for expansion, max_step_expansion > 1.0.
     double max_step_expansion = 10;

     bool is_silent = false;

     // The one dimensional function that the line search algorithm
     // minimizes.
     LineSearchFunction* function = nullptr;
   };

   // Result of the line search.
   struct Summary {
     bool success = false;
     FunctionSample optimal_point;
     int num_function_evaluations = 0;
     int num_gradient_evaluations = 0;
     int num_iterations = 0;
     // Cumulative time spent evaluating the value of the cost function across
     // all iterations.
     double cost_evaluation_time_in_seconds = 0.0;
     // Cumulative time spent evaluating the gradient of the cost function across
     // all iterations.
     double gradient_evaluation_time_in_seconds = 0.0;
     // Cumulative time spent minimizing the interpolating polynomial to compute
     // the next candidate step size across all iterations.
     double polynomial_minimization_time_in_seconds = 0.0;
     double total_time_in_seconds = 0.0;
     std::string error;
   };

   explicit LineSearch(const LineSearch::Options& options);
   virtual ~LineSearch() {}

   static LineSearch* Create(const LineSearchType line_search_type,
                             const LineSearch::Options& options,
                             std::string* error);

   // Perform the line search.
   //
   // step_size_estimate must be a positive number.
   //
   // initial_cost and initial_gradient are the values and gradient of
   // the function at zero.
   // summary must not be null and will contain the result of the line
   // search.
   //
   // Summary::success is true if a non-zero step size is found.
   void Search(double step_size_estimate,
               double initial_cost,
               double initial_gradient,
               Summary* summary) const;
   double InterpolatingPolynomialMinimizingStepSize(
       const LineSearchInterpolationType& interpolation_type,
       const FunctionSample& lowerbound_sample,
       const FunctionSample& previous_sample,
       const FunctionSample& current_sample,
       const double min_step_size,
       const double max_step_size) const;

  protected:
   const LineSearch::Options& options() const { return options_; }

  private:
   virtual void DoSearch(double step_size_estimate,
                         double initial_cost,
                         double initial_gradient,
                         Summary* summary) const = 0;

  private:
   LineSearch::Options options_;
 };

 // An object used by the line search to access the function values
 // and gradient of the one dimensional function being optimized.
 //
 // In practice, this object provides access to the objective
 // function value and the directional derivative of the underlying
 // optimization problem along a specific search direction.
 class LineSearchFunction {
  public:
   explicit LineSearchFunction(Evaluator* evaluator);
   void Init(const Vector& position, const Vector& direction);

   // Evaluate the line search objective
   //
   //   f(x) = p(position + x * direction)
   //
   // Where, p is the objective function of the general optimization
   // problem.
   //
   // evaluate_gradient controls whether the gradient will be evaluated
   // or not.
   //
   // On return output->*_is_valid indicate indicate whether the
   // corresponding fields have numerically valid values or not.
   void Evaluate(double x, bool evaluate_gradient, FunctionSample* output);

   double DirectionInfinityNorm() const;

   // Resets to now, the start point for the results from TimeStatistics().
   void ResetTimeStatistics();
   void TimeStatistics(double* cost_evaluation_time_in_seconds,
                       double* gradient_evaluation_time_in_seconds) const;
   const Vector& position() const { return position_; }
   const Vector& direction() const { return direction_; }

  private:
   Evaluator* evaluator_;
   Vector position_;
   Vector direction_;

   // scaled_direction = x * direction_;
   Vector scaled_direction_;

   // We may not exclusively own the evaluator (e.g. in the Trust Region
   // minimizer), hence we need to save the initial evaluation durations for the
   // value & gradient to accurately determine the duration of the evaluations
   // we invoked.  These are reset by a call to ResetTimeStatistics().
   double initial_evaluator_residual_time_in_seconds;
   double initial_evaluator_jacobian_time_in_seconds;
 };

 // Backtracking and interpolation based Armijo line search. This
 // implementation is based on the Armijo line search that ships in the
 // minFunc package by Mark Schmidt.
 //
 // For more details: http://www.di.ens.fr/~mschmidt/Software/minFunc.html
 class ArmijoLineSearch : public LineSearch {
  public:
   explicit ArmijoLineSearch(const LineSearch::Options& options);
   virtual ~ArmijoLineSearch() {}

  private:
   virtual void DoSearch(double step_size_estimate,
                         double initial_cost,
                         double initial_gradient,
                         Summary* summary) const;
 };

 // Bracketing / Zoom Strong Wolfe condition line search.  This implementation
 // is based on the pseudo-code algorithm presented in Nocedal & Wright [1]
 // (p60-61) with inspiration from the WolfeLineSearch which ships with the
 // minFunc package by Mark Schmidt [2].
 //
 // [1] Nocedal J., Wright S., Numerical Optimization, 2nd Ed., Springer, 1999.
 // [2] http://www.di.ens.fr/~mschmidt/Software/minFunc.html.
 class WolfeLineSearch : public LineSearch {
  public:
   explicit WolfeLineSearch(const LineSearch::Options& options);
   virtual ~WolfeLineSearch() {}

   // Returns true iff either a valid point, or valid bracket are found.
   bool BracketingPhase(const FunctionSample& initial_position,
                        const double step_size_estimate,
                        FunctionSample* bracket_low,
                        FunctionSample* bracket_high,
                        bool* perform_zoom_search,
                        Summary* summary) const;
   // Returns true iff final_line_sample satisfies strong Wolfe conditions.
   bool ZoomPhase(const FunctionSample& initial_position,
                  FunctionSample bracket_low,
                  FunctionSample bracket_high,
                  FunctionSample* solution,
                  Summary* summary) const;

  private:
   virtual void DoSearch(double step_size_estimate,
                         double initial_cost,
                         double initial_gradient,
                         Summary* summary) const;
 };

 }  // namespace internal
 }  // namespace ceres

 #endif  // CERES_INTERNAL_LINE_SEARCH_H_
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2015 Google Inc. All rights reserved.
	// http://ceres-solver.org/
	//
	// Redistribution and use in source and binary forms, with or without
	// modification, are permitted provided that the following conditions are met:
	//
	// * Redistributions of source code must retain the above copyright notice,
	// this list of conditions and the following disclaimer.
	// * Redistributions in binary form must reproduce the above copyright notice,
	// this list of conditions and the following disclaimer in the documentation
	// and/or other materials provided with the distribution.
	// * Neither the name of Google Inc. nor the names of its contributors may be
	// used to endorse or promote products derived from this software without
	// specific prior written permission.
	//
	// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
	// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
	// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
	// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
	// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
	// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
	// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
	// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
	// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
	// POSSIBILITY OF SUCH DAMAGE.
	//
	// Author: sameeragarwal@google.com (Sameer Agarwal)
	//
	// Interface for and implementation of various Line search algorithms.

	#ifndef CERES_INTERNAL_LINE_SEARCH_H_
	#define CERES_INTERNAL_LINE_SEARCH_H_

	#include <string>
	#include <vector>
	#include "ceres/function_sample.h"
	#include "ceres/internal/eigen.h"
	#include "ceres/internal/port.h"
	#include "ceres/types.h"

	namespace ceres {
	namespace internal {

	class Evaluator;
	class LineSearchFunction;

	// Line search is another name for a one dimensional optimization
	// algorithm. The name "line search" comes from the fact one
	// dimensional optimization problems that arise as subproblems of
	// general multidimensional optimization problems.
	//
	// While finding the exact minimum of a one dimensional function is
	// hard, instances of LineSearch find a point that satisfies a
	// sufficient decrease condition. Depending on the particular
	// condition used, we get a variety of different line search
	// algorithms, e.g., Armijo, Wolfe etc.
	class LineSearch {
	public:
	struct Summary;

	struct Options {
	// Degree of the polynomial used to approximate the objective
	// function.
	LineSearchInterpolationType interpolation_type = CUBIC;

	// Armijo and Wolfe line search parameters.

	// Solving the line search problem exactly is computationally
	// prohibitive. Fortunately, line search based optimization
	// algorithms can still guarantee convergence if instead of an
	// exact solution, the line search algorithm returns a solution
	// which decreases the value of the objective function
	// sufficiently. More precisely, we are looking for a step_size
	// s.t.
	//
	// f(step_size) <= f(0) + sufficient_decrease * f'(0) * step_size
	double sufficient_decrease = 1e-4;

	// In each iteration of the Armijo / Wolfe line search,
	//
	// new_step_size >= max_step_contraction * step_size
	//
	// Note that by definition, for contraction:
	//
	// 0 < max_step_contraction < min_step_contraction < 1
	//
	double max_step_contraction = 1e-3;

	// In each iteration of the Armijo / Wolfe line search,
	//
	// new_step_size <= min_step_contraction * step_size
	// Note that by definition, for contraction:
	//
	// 0 < max_step_contraction < min_step_contraction < 1
	//
	double min_step_contraction = 0.9;

	// If during the line search, the step_size falls below this
	// value, it is truncated to zero.
	double min_step_size = 1e-9;

	// Maximum number of trial step size iterations during each line search,
	// if a step size satisfying the search conditions cannot be found within
	// this number of trials, the line search will terminate.
	int max_num_iterations = 20;

	// Wolfe-specific line search parameters.

	// The strong Wolfe conditions consist of the Armijo sufficient
	// decrease condition, and an additional requirement that the
	// step-size be chosen s.t. the _magnitude_ ('strong' Wolfe
	// conditions) of the gradient along the search direction
	// decreases sufficiently. Precisely, this second condition
	// is that we seek a step_size s.t.
	//
	// \|f'(step_size)\| <= sufficient_curvature_decrease * \|f'(0)\|
	//
	// Where f() is the line search objective and f'() is the derivative
	// of f w.r.t step_size (d f / d step_size).
	double sufficient_curvature_decrease = 0.9;

	// During the bracketing phase of the Wolfe search, the step size is
	// increased until either a point satisfying the Wolfe conditions is
	// found, or an upper bound for a bracket containing a point satisfying
	// the conditions is found. Precisely, at each iteration of the
	// expansion:
	//
	// new_step_size <= max_step_expansion * step_size.
	//
	// By definition for expansion, max_step_expansion > 1.0.
	double max_step_expansion = 10;

	bool is_silent = false;

	// The one dimensional function that the line search algorithm
	// minimizes.
	LineSearchFunction* function = nullptr;
	};

	// Result of the line search.
	struct Summary {
	bool success = false;
	FunctionSample optimal_point;
	int num_function_evaluations = 0;
	int num_gradient_evaluations = 0;
	int num_iterations = 0;
	// Cumulative time spent evaluating the value of the cost function across
	// all iterations.
	double cost_evaluation_time_in_seconds = 0.0;
	// Cumulative time spent evaluating the gradient of the cost function across
	// all iterations.
	double gradient_evaluation_time_in_seconds = 0.0;
	// Cumulative time spent minimizing the interpolating polynomial to compute
	// the next candidate step size across all iterations.
	double polynomial_minimization_time_in_seconds = 0.0;
	double total_time_in_seconds = 0.0;
	std::string error;
	};

	explicit LineSearch(const LineSearch::Options& options);
	virtual ~LineSearch() {}

	static LineSearch* Create(const LineSearchType line_search_type,
	const LineSearch::Options& options,
	std::string* error);

	// Perform the line search.
	//
	// step_size_estimate must be a positive number.
	//
	// initial_cost and initial_gradient are the values and gradient of
	// the function at zero.
	// summary must not be null and will contain the result of the line
	// search.
	//
	// Summary::success is true if a non-zero step size is found.
	void Search(double step_size_estimate,
	double initial_cost,
	double initial_gradient,
	Summary* summary) const;
	double InterpolatingPolynomialMinimizingStepSize(
	const LineSearchInterpolationType& interpolation_type,
	const FunctionSample& lowerbound_sample,
	const FunctionSample& previous_sample,
	const FunctionSample& current_sample,
	const double min_step_size,
	const double max_step_size) const;

	protected:
	const LineSearch::Options& options() const { return options_; }

	private:
	virtual void DoSearch(double step_size_estimate,
	double initial_cost,
	double initial_gradient,
	Summary* summary) const = 0;

	private:
	LineSearch::Options options_;
	};

	// An object used by the line search to access the function values
	// and gradient of the one dimensional function being optimized.
	//
	// In practice, this object provides access to the objective
	// function value and the directional derivative of the underlying
	// optimization problem along a specific search direction.
	class LineSearchFunction {
	public:
	explicit LineSearchFunction(Evaluator* evaluator);
	void Init(const Vector& position, const Vector& direction);

	// Evaluate the line search objective
	//
	// f(x) = p(position + x * direction)
	//
	// Where, p is the objective function of the general optimization
	// problem.
	//
	// evaluate_gradient controls whether the gradient will be evaluated
	// or not.
	//
	// On return output->*_is_valid indicate indicate whether the
	// corresponding fields have numerically valid values or not.
	void Evaluate(double x, bool evaluate_gradient, FunctionSample* output);

	double DirectionInfinityNorm() const;

	// Resets to now, the start point for the results from TimeStatistics().
	void ResetTimeStatistics();
	void TimeStatistics(double* cost_evaluation_time_in_seconds,
	double* gradient_evaluation_time_in_seconds) const;
	const Vector& position() const { return position_; }
	const Vector& direction() const { return direction_; }

	private:
	Evaluator* evaluator_;
	Vector position_;
	Vector direction_;

	// scaled_direction = x * direction_;
	Vector scaled_direction_;

	// We may not exclusively own the evaluator (e.g. in the Trust Region
	// minimizer), hence we need to save the initial evaluation durations for the
	// value & gradient to accurately determine the duration of the evaluations
	// we invoked. These are reset by a call to ResetTimeStatistics().
	double initial_evaluator_residual_time_in_seconds;
	double initial_evaluator_jacobian_time_in_seconds;
	};

	// Backtracking and interpolation based Armijo line search. This
	// implementation is based on the Armijo line search that ships in the
	// minFunc package by Mark Schmidt.
	//
	// For more details: http://www.di.ens.fr/~mschmidt/Software/minFunc.html
	class ArmijoLineSearch : public LineSearch {
	public:
	explicit ArmijoLineSearch(const LineSearch::Options& options);
	virtual ~ArmijoLineSearch() {}

	private:
	virtual void DoSearch(double step_size_estimate,
	double initial_cost,
	double initial_gradient,
	Summary* summary) const;
	};

	// Bracketing / Zoom Strong Wolfe condition line search. This implementation
	// is based on the pseudo-code algorithm presented in Nocedal & Wright [1]
	// (p60-61) with inspiration from the WolfeLineSearch which ships with the
	// minFunc package by Mark Schmidt [2].
	//
	// [1] Nocedal J., Wright S., Numerical Optimization, 2nd Ed., Springer, 1999.
	// [2] http://www.di.ens.fr/~mschmidt/Software/minFunc.html.
	class WolfeLineSearch : public LineSearch {
	public:
	explicit WolfeLineSearch(const LineSearch::Options& options);
	virtual ~WolfeLineSearch() {}

	// Returns true iff either a valid point, or valid bracket are found.
	bool BracketingPhase(const FunctionSample& initial_position,
	const double step_size_estimate,
	FunctionSample* bracket_low,
	FunctionSample* bracket_high,
	bool* perform_zoom_search,
	Summary* summary) const;
	// Returns true iff final_line_sample satisfies strong Wolfe conditions.
	bool ZoomPhase(const FunctionSample& initial_position,
	FunctionSample bracket_low,
	FunctionSample bracket_high,
	FunctionSample* solution,
	Summary* summary) const;

	private:
	virtual void DoSearch(double step_size_estimate,
	double initial_cost,
	double initial_gradient,
	Summary* summary) const;
	};

	} // namespace internal
	} // namespace ceres

	#endif // CERES_INTERNAL_LINE_SEARCH_H_