// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
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// 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
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// 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)
//
// Generic loop for line search based optimization algorithms.
//
// This is primarily inpsired by the minFunc packaged written by Mark
// Schmidt.
//
// http://www.di.ens.fr/~mschmidt/Software/minFunc.html
//
// For details on the theory and implementation see "Numerical
// Optimization" by Nocedal & Wright.
#include "ceres/line_search_minimizer.h"
#include <algorithm>
#include <cstdlib>
#include <cmath>
#include <string>
#include <vector>
#include "Eigen/Dense"
#include "ceres/array_utils.h"
#include "ceres/evaluator.h"
#include "ceres/internal/eigen.h"
#include "ceres/internal/port.h"
#include "ceres/internal/scoped_ptr.h"
#include "ceres/line_search.h"
#include "ceres/line_search_direction.h"
#include "ceres/stringprintf.h"
#include "ceres/types.h"
#include "ceres/wall_time.h"
#include "glog/logging.h"
namespace ceres {
namespace internal {
namespace {
bool EvaluateGradientNorms(Evaluator* evaluator,
const Vector& x,
LineSearchMinimizer::State* state,
std::string* message) {
Vector negative_gradient = -state->gradient;
Vector projected_gradient_step(x.size());
if (!evaluator->Plus(
x.data(), negative_gradient.data(), projected_gradient_step.data())) {
*message = "projected_gradient_step = Plus(x, -gradient) failed.";
return false;
}
state->gradient_squared_norm = (x - projected_gradient_step).squaredNorm();
state->gradient_max_norm =
(x - projected_gradient_step).lpNorm<Eigen::Infinity>();
return true;
}
} // namespace
void LineSearchMinimizer::Minimize(const Minimizer::Options& options,
double* parameters,
Solver::Summary* summary) {
const bool is_not_silent = !options.is_silent;
double start_time = WallTimeInSeconds();
double iteration_start_time = start_time;
Evaluator* evaluator = CHECK_NOTNULL(options.evaluator.get());
const int num_parameters = evaluator->NumParameters();
const int num_effective_parameters = evaluator->NumEffectiveParameters();
summary->termination_type = NO_CONVERGENCE;
summary->num_successful_steps = 0;
summary->num_unsuccessful_steps = 0;
VectorRef x(parameters, num_parameters);
State current_state(num_parameters, num_effective_parameters);
State previous_state(num_parameters, num_effective_parameters);
IterationSummary iteration_summary;
iteration_summary.iteration = 0;
iteration_summary.step_is_valid = false;
iteration_summary.step_is_successful = false;
iteration_summary.cost_change = 0.0;
iteration_summary.gradient_max_norm = 0.0;
iteration_summary.gradient_norm = 0.0;
iteration_summary.step_norm = 0.0;
iteration_summary.linear_solver_iterations = 0;
iteration_summary.step_solver_time_in_seconds = 0;
// Do initial cost and gradient evaluation.
if (!evaluator->Evaluate(x.data(),
&(current_state.cost),
NULL,
current_state.gradient.data(),
NULL)) {
summary->termination_type = FAILURE;
summary->message = "Initial cost and jacobian evaluation failed.";
LOG_IF(WARNING, is_not_silent) << "Terminating: " << summary->message;
return;
}
if (!EvaluateGradientNorms(evaluator, x, ¤t_state, &summary->message)) {
summary->termination_type = FAILURE;
summary->message = "Initial cost and jacobian evaluation failed. "
"More details: " + summary->message;
LOG_IF(WARNING, is_not_silent) << "Terminating: " << summary->message;
return;
}
summary->initial_cost = current_state.cost + summary->fixed_cost;
iteration_summary.cost = current_state.cost + summary->fixed_cost;
iteration_summary.gradient_norm = sqrt(current_state.gradient_squared_norm);
iteration_summary.gradient_max_norm = current_state.gradient_max_norm;
if (iteration_summary.gradient_max_norm <= options.gradient_tolerance) {
summary->message = StringPrintf("Gradient tolerance reached. "
"Gradient max norm: %e <= %e",
iteration_summary.gradient_max_norm,
options.gradient_tolerance);
summary->termination_type = CONVERGENCE;
VLOG_IF(1, is_not_silent) << "Terminating: " << summary->message;
return;
}
iteration_summary.iteration_time_in_seconds =
WallTimeInSeconds() - iteration_start_time;
iteration_summary.cumulative_time_in_seconds =
WallTimeInSeconds() - start_time
+ summary->preprocessor_time_in_seconds;
summary->iterations.push_back(iteration_summary);
LineSearchDirection::Options line_search_direction_options;
line_search_direction_options.num_parameters = num_effective_parameters;
line_search_direction_options.type = options.line_search_direction_type;
line_search_direction_options.nonlinear_conjugate_gradient_type =
options.nonlinear_conjugate_gradient_type;
line_search_direction_options.max_lbfgs_rank = options.max_lbfgs_rank;
line_search_direction_options.use_approximate_eigenvalue_bfgs_scaling =
options.use_approximate_eigenvalue_bfgs_scaling;
scoped_ptr<LineSearchDirection> line_search_direction(
LineSearchDirection::Create(line_search_direction_options));
LineSearchFunction line_search_function(evaluator);
LineSearch::Options line_search_options;
line_search_options.interpolation_type =
options.line_search_interpolation_type;
line_search_options.min_step_size = options.min_line_search_step_size;
line_search_options.sufficient_decrease =
options.line_search_sufficient_function_decrease;
line_search_options.max_step_contraction =
options.max_line_search_step_contraction;
line_search_options.min_step_contraction =
options.min_line_search_step_contraction;
line_search_options.max_num_iterations =
options.max_num_line_search_step_size_iterations;
line_search_options.sufficient_curvature_decrease =
options.line_search_sufficient_curvature_decrease;
line_search_options.max_step_expansion =
options.max_line_search_step_expansion;
line_search_options.is_silent = options.is_silent;
line_search_options.function = &line_search_function;
scoped_ptr<LineSearch>
line_search(LineSearch::Create(options.line_search_type,
line_search_options,
&summary->message));
if (line_search.get() == NULL) {
summary->termination_type = FAILURE;
LOG_IF(ERROR, is_not_silent) << "Terminating: " << summary->message;
return;
}
LineSearch::Summary line_search_summary;
int num_line_search_direction_restarts = 0;
while (true) {
if (!RunCallbacks(options, iteration_summary, summary)) {
break;
}
iteration_start_time = WallTimeInSeconds();
if (iteration_summary.iteration >= options.max_num_iterations) {
summary->message = "Maximum number of iterations reached.";
summary->termination_type = NO_CONVERGENCE;
VLOG_IF(1, is_not_silent) << "Terminating: " << summary->message;
break;
}
const double total_solver_time = iteration_start_time - start_time +
summary->preprocessor_time_in_seconds;
if (total_solver_time >= options.max_solver_time_in_seconds) {
summary->message = "Maximum solver time reached.";
summary->termination_type = NO_CONVERGENCE;
VLOG_IF(1, is_not_silent) << "Terminating: " << summary->message;
break;
}
iteration_summary = IterationSummary();
iteration_summary.iteration = summary->iterations.back().iteration + 1;
iteration_summary.step_is_valid = false;
iteration_summary.step_is_successful = false;
bool line_search_status = true;
if (iteration_summary.iteration == 1) {
current_state.search_direction = -current_state.gradient;
} else {
line_search_status = line_search_direction->NextDirection(
previous_state,
current_state,
¤t_state.search_direction);
}
if (!line_search_status &&
num_line_search_direction_restarts >=
options.max_num_line_search_direction_restarts) {
// Line search direction failed to generate a new direction, and we
// have already reached our specified maximum number of restarts,
// terminate optimization.
summary->message =
StringPrintf("Line search direction failure: specified "
"max_num_line_search_direction_restarts: %d reached.",
options.max_num_line_search_direction_restarts);
summary->termination_type = FAILURE;
LOG_IF(WARNING, is_not_silent) << "Terminating: " << summary->message;
break;
} else if (!line_search_status) {
// Restart line search direction with gradient descent on first iteration
// as we have not yet reached our maximum number of restarts.
CHECK_LT(num_line_search_direction_restarts,
options.max_num_line_search_direction_restarts);
++num_line_search_direction_restarts;
LOG_IF(WARNING, is_not_silent)
<< "Line search direction algorithm: "
<< LineSearchDirectionTypeToString(
options.line_search_direction_type)
<< ", failed to produce a valid new direction at "
<< "iteration: " << iteration_summary.iteration
<< ". Restarting, number of restarts: "
<< num_line_search_direction_restarts << " / "
<< options.max_num_line_search_direction_restarts
<< " [max].";
line_search_direction.reset(
LineSearchDirection::Create(line_search_direction_options));
current_state.search_direction = -current_state.gradient;
}
line_search_function.Init(x, current_state.search_direction);
current_state.directional_derivative =
current_state.gradient.dot(current_state.search_direction);
// TODO(sameeragarwal): Refactor this into its own object and add
// explanations for the various choices.
//
// Note that we use !line_search_status to ensure that we treat cases when
// we restarted the line search direction equivalently to the first
// iteration.
const double initial_step_size =
(iteration_summary.iteration == 1 || !line_search_status)
? std::min(1.0, 1.0 / current_state.gradient_max_norm)
: std::min(1.0, 2.0 * (current_state.cost - previous_state.cost) /
current_state.directional_derivative);
// By definition, we should only ever go forwards along the specified search
// direction in a line search, most likely cause for this being violated
// would be a numerical failure in the line search direction calculation.
if (initial_step_size < 0.0) {
summary->message =
StringPrintf("Numerical failure in line search, initial_step_size is "
"negative: %.5e, directional_derivative: %.5e, "
"(current_cost - previous_cost): %.5e",
initial_step_size, current_state.directional_derivative,
(current_state.cost - previous_state.cost));
summary->termination_type = FAILURE;
LOG_IF(WARNING, is_not_silent) << "Terminating: " << summary->message;
break;
}
line_search->Search(initial_step_size,
current_state.cost,
current_state.directional_derivative,
&line_search_summary);
if (!line_search_summary.success) {
summary->message =
StringPrintf("Numerical failure in line search, failed to find "
"a valid step size, (did not run out of iterations) "
"using initial_step_size: %.5e, initial_cost: %.5e, "
"initial_gradient: %.5e.",
initial_step_size, current_state.cost,
current_state.directional_derivative);
LOG_IF(WARNING, is_not_silent) << "Terminating: " << summary->message;
summary->termination_type = FAILURE;
break;
}
const FunctionSample& optimal_point = line_search_summary.optimal_point;
CHECK(optimal_point.vector_x_is_valid)
<< "Congratulations, you found a bug in Ceres. Please report it.";
current_state.step_size = optimal_point.x;
previous_state = current_state;
iteration_summary.step_solver_time_in_seconds =
WallTimeInSeconds() - iteration_start_time;
if (optimal_point.vector_gradient_is_valid) {
current_state.cost = optimal_point.value;
current_state.gradient = optimal_point.vector_gradient;
} else {
if (!evaluator->Evaluate(optimal_point.vector_x.data(),
&(current_state.cost),
NULL,
current_state.gradient.data(),
NULL)) {
summary->termination_type = FAILURE;
summary->message = "Cost and jacobian evaluation failed.";
LOG_IF(WARNING, is_not_silent) << "Terminating: " << summary->message;
return;
}
}
if (!EvaluateGradientNorms(evaluator,
optimal_point.vector_x,
¤t_state,
&summary->message)) {
summary->termination_type = FAILURE;
summary->message =
"Step failed to evaluate. This should not happen as the step was "
"valid when it was selected by the line search. More details: " +
summary->message;
LOG_IF(WARNING, is_not_silent) << "Terminating: " << summary->message;
break;
}
// Compute the norm of the step in the ambient space.
iteration_summary.step_norm = (optimal_point.vector_x - x).norm();
const double x_norm = x.norm();
x = optimal_point.vector_x;
iteration_summary.gradient_max_norm = current_state.gradient_max_norm;
iteration_summary.gradient_norm = sqrt(current_state.gradient_squared_norm);
iteration_summary.cost_change = previous_state.cost - current_state.cost;
iteration_summary.cost = current_state.cost + summary->fixed_cost;
iteration_summary.step_is_valid = true;
iteration_summary.step_is_successful = true;
iteration_summary.step_size = current_state.step_size;
iteration_summary.line_search_function_evaluations =
line_search_summary.num_function_evaluations;
iteration_summary.line_search_gradient_evaluations =
line_search_summary.num_gradient_evaluations;
iteration_summary.line_search_iterations =
line_search_summary.num_iterations;
iteration_summary.iteration_time_in_seconds =
WallTimeInSeconds() - iteration_start_time;
iteration_summary.cumulative_time_in_seconds =
WallTimeInSeconds() - start_time
+ summary->preprocessor_time_in_seconds;
summary->iterations.push_back(iteration_summary);
// Iterations inside the line search algorithm are considered
// 'steps' in the broader context, to distinguish these inner
// iterations from from the outer iterations of the line search
// minimizer. The number of line search steps is the total number
// of inner line search iterations (or steps) across the entire
// minimization.
summary->num_line_search_steps += line_search_summary.num_iterations;
summary->line_search_cost_evaluation_time_in_seconds +=
line_search_summary.cost_evaluation_time_in_seconds;
summary->line_search_gradient_evaluation_time_in_seconds +=
line_search_summary.gradient_evaluation_time_in_seconds;
summary->line_search_polynomial_minimization_time_in_seconds +=
line_search_summary.polynomial_minimization_time_in_seconds;
summary->line_search_total_time_in_seconds +=
line_search_summary.total_time_in_seconds;
++summary->num_successful_steps;
const double step_size_tolerance = options.parameter_tolerance *
(x_norm + options.parameter_tolerance);
if (iteration_summary.step_norm <= step_size_tolerance) {
summary->message =
StringPrintf("Parameter tolerance reached. "
"Relative step_norm: %e <= %e.",
(iteration_summary.step_norm /
(x_norm + options.parameter_tolerance)),
options.parameter_tolerance);
summary->termination_type = CONVERGENCE;
VLOG_IF(1, is_not_silent) << "Terminating: " << summary->message;
return;
}
if (iteration_summary.gradient_max_norm <= options.gradient_tolerance) {
summary->message = StringPrintf("Gradient tolerance reached. "
"Gradient max norm: %e <= %e",
iteration_summary.gradient_max_norm,
options.gradient_tolerance);
summary->termination_type = CONVERGENCE;
VLOG_IF(1, is_not_silent) << "Terminating: " << summary->message;
break;
}
const double absolute_function_tolerance =
options.function_tolerance * previous_state.cost;
if (fabs(iteration_summary.cost_change) <= absolute_function_tolerance) {
summary->message =
StringPrintf("Function tolerance reached. "
"|cost_change|/cost: %e <= %e",
fabs(iteration_summary.cost_change) /
previous_state.cost,
options.function_tolerance);
summary->termination_type = CONVERGENCE;
VLOG_IF(1, is_not_silent) << "Terminating: " << summary->message;
break;
}
}
}
} // namespace internal
} // namespace ceres