Squashed 'third_party/ceres/' content from commit e51e9b4
Change-Id: I763587619d57e594d3fa158dc3a7fe0b89a1743b
git-subtree-dir: third_party/ceres
git-subtree-split: e51e9b46f6ca88ab8b2266d0e362771db6d98067
diff --git a/internal/ceres/line_search.cc b/internal/ceres/line_search.cc
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+// 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)
+
+#include "ceres/line_search.h"
+
+#include <algorithm>
+#include <cmath>
+#include <iomanip>
+#include <iostream> // NOLINT
+
+#include "ceres/evaluator.h"
+#include "ceres/function_sample.h"
+#include "ceres/internal/eigen.h"
+#include "ceres/map_util.h"
+#include "ceres/polynomial.h"
+#include "ceres/stringprintf.h"
+#include "ceres/wall_time.h"
+#include "glog/logging.h"
+
+namespace ceres {
+namespace internal {
+
+using std::map;
+using std::ostream;
+using std::string;
+using std::vector;
+
+namespace {
+// Precision used for floating point values in error message output.
+const int kErrorMessageNumericPrecision = 8;
+} // namespace
+
+ostream& operator<<(ostream &os, const FunctionSample& sample);
+
+// Convenience stream operator for pushing FunctionSamples into log messages.
+ostream& operator<<(ostream &os, const FunctionSample& sample) {
+ os << sample.ToDebugString();
+ return os;
+}
+
+LineSearch::LineSearch(const LineSearch::Options& options)
+ : options_(options) {}
+
+LineSearch* LineSearch::Create(const LineSearchType line_search_type,
+ const LineSearch::Options& options,
+ string* error) {
+ LineSearch* line_search = NULL;
+ switch (line_search_type) {
+ case ceres::ARMIJO:
+ line_search = new ArmijoLineSearch(options);
+ break;
+ case ceres::WOLFE:
+ line_search = new WolfeLineSearch(options);
+ break;
+ default:
+ *error = string("Invalid line search algorithm type: ") +
+ LineSearchTypeToString(line_search_type) +
+ string(", unable to create line search.");
+ return NULL;
+ }
+ return line_search;
+}
+
+LineSearchFunction::LineSearchFunction(Evaluator* evaluator)
+ : evaluator_(evaluator),
+ position_(evaluator->NumParameters()),
+ direction_(evaluator->NumEffectiveParameters()),
+ scaled_direction_(evaluator->NumEffectiveParameters()),
+ initial_evaluator_residual_time_in_seconds(0.0),
+ initial_evaluator_jacobian_time_in_seconds(0.0) {}
+
+void LineSearchFunction::Init(const Vector& position,
+ const Vector& direction) {
+ position_ = position;
+ direction_ = direction;
+}
+
+void LineSearchFunction::Evaluate(const double x,
+ const bool evaluate_gradient,
+ FunctionSample* output) {
+ output->x = x;
+ output->vector_x_is_valid = false;
+ output->value_is_valid = false;
+ output->gradient_is_valid = false;
+ output->vector_gradient_is_valid = false;
+
+ scaled_direction_ = output->x * direction_;
+ output->vector_x.resize(position_.rows(), 1);
+ if (!evaluator_->Plus(position_.data(),
+ scaled_direction_.data(),
+ output->vector_x.data())) {
+ return;
+ }
+ output->vector_x_is_valid = true;
+
+ double* gradient = NULL;
+ if (evaluate_gradient) {
+ output->vector_gradient.resize(direction_.rows(), 1);
+ gradient = output->vector_gradient.data();
+ }
+ const bool eval_status = evaluator_->Evaluate(
+ output->vector_x.data(), &(output->value), NULL, gradient, NULL);
+
+ if (!eval_status || !std::isfinite(output->value)) {
+ return;
+ }
+
+ output->value_is_valid = true;
+ if (!evaluate_gradient) {
+ return;
+ }
+
+ output->gradient = direction_.dot(output->vector_gradient);
+ if (!std::isfinite(output->gradient)) {
+ return;
+ }
+
+ output->gradient_is_valid = true;
+ output->vector_gradient_is_valid = true;
+}
+
+double LineSearchFunction::DirectionInfinityNorm() const {
+ return direction_.lpNorm<Eigen::Infinity>();
+}
+
+void LineSearchFunction::ResetTimeStatistics() {
+ const map<string, CallStatistics> evaluator_statistics =
+ evaluator_->Statistics();
+
+ initial_evaluator_residual_time_in_seconds =
+ FindWithDefault(
+ evaluator_statistics, "Evaluator::Residual", CallStatistics())
+ .time;
+ initial_evaluator_jacobian_time_in_seconds =
+ FindWithDefault(
+ evaluator_statistics, "Evaluator::Jacobian", CallStatistics())
+ .time;
+}
+
+void LineSearchFunction::TimeStatistics(
+ double* cost_evaluation_time_in_seconds,
+ double* gradient_evaluation_time_in_seconds) const {
+ const map<string, CallStatistics> evaluator_time_statistics =
+ evaluator_->Statistics();
+ *cost_evaluation_time_in_seconds =
+ FindWithDefault(
+ evaluator_time_statistics, "Evaluator::Residual", CallStatistics())
+ .time -
+ initial_evaluator_residual_time_in_seconds;
+ // Strictly speaking this will slightly underestimate the time spent
+ // evaluating the gradient of the line search univariate cost function as it
+ // does not count the time spent performing the dot product with the direction
+ // vector. However, this will typically be small by comparison, and also
+ // allows direct subtraction of the timing information from the totals for
+ // the evaluator returned in the solver summary.
+ *gradient_evaluation_time_in_seconds =
+ FindWithDefault(
+ evaluator_time_statistics, "Evaluator::Jacobian", CallStatistics())
+ .time -
+ initial_evaluator_jacobian_time_in_seconds;
+}
+
+void LineSearch::Search(double step_size_estimate,
+ double initial_cost,
+ double initial_gradient,
+ Summary* summary) const {
+ const double start_time = WallTimeInSeconds();
+ CHECK(summary != nullptr);
+ *summary = LineSearch::Summary();
+
+ summary->cost_evaluation_time_in_seconds = 0.0;
+ summary->gradient_evaluation_time_in_seconds = 0.0;
+ summary->polynomial_minimization_time_in_seconds = 0.0;
+ options().function->ResetTimeStatistics();
+ this->DoSearch(step_size_estimate, initial_cost, initial_gradient, summary);
+ options().function->
+ TimeStatistics(&summary->cost_evaluation_time_in_seconds,
+ &summary->gradient_evaluation_time_in_seconds);
+
+ summary->total_time_in_seconds = WallTimeInSeconds() - start_time;
+}
+
+// Returns step_size \in [min_step_size, max_step_size] which minimizes the
+// polynomial of degree defined by interpolation_type which interpolates all
+// of the provided samples with valid values.
+double LineSearch::InterpolatingPolynomialMinimizingStepSize(
+ const LineSearchInterpolationType& interpolation_type,
+ const FunctionSample& lowerbound,
+ const FunctionSample& previous,
+ const FunctionSample& current,
+ const double min_step_size,
+ const double max_step_size) const {
+ if (!current.value_is_valid ||
+ (interpolation_type == BISECTION &&
+ max_step_size <= current.x)) {
+ // Either: sample is invalid; or we are using BISECTION and contracting
+ // the step size.
+ return std::min(std::max(current.x * 0.5, min_step_size), max_step_size);
+ } else if (interpolation_type == BISECTION) {
+ CHECK_GT(max_step_size, current.x);
+ // We are expanding the search (during a Wolfe bracketing phase) using
+ // BISECTION interpolation. Using BISECTION when trying to expand is
+ // strictly speaking an oxymoron, but we define this to mean always taking
+ // the maximum step size so that the Armijo & Wolfe implementations are
+ // agnostic to the interpolation type.
+ return max_step_size;
+ }
+ // Only check if lower-bound is valid here, where it is required
+ // to avoid replicating current.value_is_valid == false
+ // behaviour in WolfeLineSearch.
+ CHECK(lowerbound.value_is_valid)
+ << std::scientific << std::setprecision(kErrorMessageNumericPrecision)
+ << "Ceres bug: lower-bound sample for interpolation is invalid, "
+ << "please contact the developers!, interpolation_type: "
+ << LineSearchInterpolationTypeToString(interpolation_type)
+ << ", lowerbound: " << lowerbound << ", previous: " << previous
+ << ", current: " << current;
+
+ // Select step size by interpolating the function and gradient values
+ // and minimizing the corresponding polynomial.
+ vector<FunctionSample> samples;
+ samples.push_back(lowerbound);
+
+ if (interpolation_type == QUADRATIC) {
+ // Two point interpolation using function values and the
+ // gradient at the lower bound.
+ samples.push_back(FunctionSample(current.x, current.value));
+
+ if (previous.value_is_valid) {
+ // Three point interpolation, using function values and the
+ // gradient at the lower bound.
+ samples.push_back(FunctionSample(previous.x, previous.value));
+ }
+ } else if (interpolation_type == CUBIC) {
+ // Two point interpolation using the function values and the gradients.
+ samples.push_back(current);
+
+ if (previous.value_is_valid) {
+ // Three point interpolation using the function values and
+ // the gradients.
+ samples.push_back(previous);
+ }
+ } else {
+ LOG(FATAL) << "Ceres bug: No handler for interpolation_type: "
+ << LineSearchInterpolationTypeToString(interpolation_type)
+ << ", please contact the developers!";
+ }
+
+ double step_size = 0.0, unused_min_value = 0.0;
+ MinimizeInterpolatingPolynomial(samples, min_step_size, max_step_size,
+ &step_size, &unused_min_value);
+ return step_size;
+}
+
+ArmijoLineSearch::ArmijoLineSearch(const LineSearch::Options& options)
+ : LineSearch(options) {}
+
+void ArmijoLineSearch::DoSearch(const double step_size_estimate,
+ const double initial_cost,
+ const double initial_gradient,
+ Summary* summary) const {
+ CHECK_GE(step_size_estimate, 0.0);
+ CHECK_GT(options().sufficient_decrease, 0.0);
+ CHECK_LT(options().sufficient_decrease, 1.0);
+ CHECK_GT(options().max_num_iterations, 0);
+ LineSearchFunction* function = options().function;
+
+ // Note initial_cost & initial_gradient are evaluated at step_size = 0,
+ // not step_size_estimate, which is our starting guess.
+ FunctionSample initial_position(0.0, initial_cost, initial_gradient);
+ initial_position.vector_x = function->position();
+ initial_position.vector_x_is_valid = true;
+
+ const double descent_direction_max_norm = function->DirectionInfinityNorm();
+ FunctionSample previous;
+ FunctionSample current;
+
+ // As the Armijo line search algorithm always uses the initial point, for
+ // which both the function value and derivative are known, when fitting a
+ // minimizing polynomial, we can fit up to a quadratic without requiring the
+ // gradient at the current query point.
+ const bool kEvaluateGradient = options().interpolation_type == CUBIC;
+
+ ++summary->num_function_evaluations;
+ if (kEvaluateGradient) {
+ ++summary->num_gradient_evaluations;
+ }
+
+ function->Evaluate(step_size_estimate, kEvaluateGradient, ¤t);
+ while (!current.value_is_valid ||
+ current.value > (initial_cost
+ + options().sufficient_decrease
+ * initial_gradient
+ * current.x)) {
+ // If current.value_is_valid is false, we treat it as if the cost at that
+ // point is not large enough to satisfy the sufficient decrease condition.
+ ++summary->num_iterations;
+ if (summary->num_iterations >= options().max_num_iterations) {
+ summary->error =
+ StringPrintf("Line search failed: Armijo failed to find a point "
+ "satisfying the sufficient decrease condition within "
+ "specified max_num_iterations: %d.",
+ options().max_num_iterations);
+ LOG_IF(WARNING, !options().is_silent) << summary->error;
+ return;
+ }
+
+ const double polynomial_minimization_start_time = WallTimeInSeconds();
+ const double step_size =
+ this->InterpolatingPolynomialMinimizingStepSize(
+ options().interpolation_type,
+ initial_position,
+ previous,
+ current,
+ (options().max_step_contraction * current.x),
+ (options().min_step_contraction * current.x));
+ summary->polynomial_minimization_time_in_seconds +=
+ (WallTimeInSeconds() - polynomial_minimization_start_time);
+
+ if (step_size * descent_direction_max_norm < options().min_step_size) {
+ summary->error =
+ StringPrintf("Line search failed: step_size too small: %.5e "
+ "with descent_direction_max_norm: %.5e.", step_size,
+ descent_direction_max_norm);
+ LOG_IF(WARNING, !options().is_silent) << summary->error;
+ return;
+ }
+
+ previous = current;
+
+ ++summary->num_function_evaluations;
+ if (kEvaluateGradient) {
+ ++summary->num_gradient_evaluations;
+ }
+
+ function->Evaluate(step_size, kEvaluateGradient, ¤t);
+ }
+
+ summary->optimal_point = current;
+ summary->success = true;
+}
+
+WolfeLineSearch::WolfeLineSearch(const LineSearch::Options& options)
+ : LineSearch(options) {}
+
+void WolfeLineSearch::DoSearch(const double step_size_estimate,
+ const double initial_cost,
+ const double initial_gradient,
+ Summary* summary) const {
+ // All parameters should have been validated by the Solver, but as
+ // invalid values would produce crazy nonsense, hard check them here.
+ CHECK_GE(step_size_estimate, 0.0);
+ CHECK_GT(options().sufficient_decrease, 0.0);
+ CHECK_GT(options().sufficient_curvature_decrease,
+ options().sufficient_decrease);
+ CHECK_LT(options().sufficient_curvature_decrease, 1.0);
+ CHECK_GT(options().max_step_expansion, 1.0);
+
+ // Note initial_cost & initial_gradient are evaluated at step_size = 0,
+ // not step_size_estimate, which is our starting guess.
+ FunctionSample initial_position(0.0, initial_cost, initial_gradient);
+ initial_position.vector_x = options().function->position();
+ initial_position.vector_x_is_valid = true;
+ bool do_zoom_search = false;
+ // Important: The high/low in bracket_high & bracket_low refer to their
+ // _function_ values, not their step sizes i.e. it is _not_ required that
+ // bracket_low.x < bracket_high.x.
+ FunctionSample solution, bracket_low, bracket_high;
+
+ // Wolfe bracketing phase: Increases step_size until either it finds a point
+ // that satisfies the (strong) Wolfe conditions, or an interval that brackets
+ // step sizes which satisfy the conditions. From Nocedal & Wright [1] p61 the
+ // interval: (step_size_{k-1}, step_size_{k}) contains step lengths satisfying
+ // the strong Wolfe conditions if one of the following conditions are met:
+ //
+ // 1. step_size_{k} violates the sufficient decrease (Armijo) condition.
+ // 2. f(step_size_{k}) >= f(step_size_{k-1}).
+ // 3. f'(step_size_{k}) >= 0.
+ //
+ // Caveat: If f(step_size_{k}) is invalid, then step_size is reduced, ignoring
+ // this special case, step_size monotonically increases during bracketing.
+ if (!this->BracketingPhase(initial_position,
+ step_size_estimate,
+ &bracket_low,
+ &bracket_high,
+ &do_zoom_search,
+ summary)) {
+ // Failed to find either a valid point, a valid bracket satisfying the Wolfe
+ // conditions, or even a step size > minimum tolerance satisfying the Armijo
+ // condition.
+ return;
+ }
+
+ if (!do_zoom_search) {
+ // Either: Bracketing phase already found a point satisfying the strong
+ // Wolfe conditions, thus no Zoom required.
+ //
+ // Or: Bracketing failed to find a valid bracket or a point satisfying the
+ // strong Wolfe conditions within max_num_iterations, or whilst searching
+ // shrank the bracket width until it was below our minimum tolerance.
+ // As these are 'artificial' constraints, and we would otherwise fail to
+ // produce a valid point when ArmijoLineSearch would succeed, we return the
+ // point with the lowest cost found thus far which satsifies the Armijo
+ // condition (but not the Wolfe conditions).
+ summary->optimal_point = bracket_low;
+ summary->success = true;
+ return;
+ }
+
+ VLOG(3) << std::scientific << std::setprecision(kErrorMessageNumericPrecision)
+ << "Starting line search zoom phase with bracket_low: "
+ << bracket_low << ", bracket_high: " << bracket_high
+ << ", bracket width: " << fabs(bracket_low.x - bracket_high.x)
+ << ", bracket abs delta cost: "
+ << fabs(bracket_low.value - bracket_high.value);
+
+ // Wolfe Zoom phase: Called when the Bracketing phase finds an interval of
+ // non-zero, finite width that should bracket step sizes which satisfy the
+ // (strong) Wolfe conditions (before finding a step size that satisfies the
+ // conditions). Zoom successively decreases the size of the interval until a
+ // step size which satisfies the Wolfe conditions is found. The interval is
+ // defined by bracket_low & bracket_high, which satisfy:
+ //
+ // 1. The interval bounded by step sizes: bracket_low.x & bracket_high.x
+ // contains step sizes that satsify the strong Wolfe conditions.
+ // 2. bracket_low.x is of all the step sizes evaluated *which satisifed the
+ // Armijo sufficient decrease condition*, the one which generated the
+ // smallest function value, i.e. bracket_low.value <
+ // f(all other steps satisfying Armijo).
+ // - Note that this does _not_ (necessarily) mean that initially
+ // bracket_low.value < bracket_high.value (although this is typical)
+ // e.g. when bracket_low = initial_position, and bracket_high is the
+ // first sample, and which does not satisfy the Armijo condition,
+ // but still has bracket_high.value < initial_position.value.
+ // 3. bracket_high is chosen after bracket_low, s.t.
+ // bracket_low.gradient * (bracket_high.x - bracket_low.x) < 0.
+ if (!this->ZoomPhase(initial_position,
+ bracket_low,
+ bracket_high,
+ &solution,
+ summary) && !solution.value_is_valid) {
+ // Failed to find a valid point (given the specified decrease parameters)
+ // within the specified bracket.
+ return;
+ }
+ // Ensure that if we ran out of iterations whilst zooming the bracket, or
+ // shrank the bracket width to < tolerance and failed to find a point which
+ // satisfies the strong Wolfe curvature condition, that we return the point
+ // amongst those found thus far, which minimizes f() and satisfies the Armijo
+ // condition.
+
+ if (!solution.value_is_valid || solution.value > bracket_low.value) {
+ summary->optimal_point = bracket_low;
+ } else {
+ summary->optimal_point = solution;
+ }
+
+ summary->success = true;
+}
+
+// Returns true if either:
+//
+// A termination condition satisfying the (strong) Wolfe bracketing conditions
+// is found:
+//
+// - A valid point, defined as a bracket of zero width [zoom not required].
+// - A valid bracket (of width > tolerance), [zoom required].
+//
+// Or, searching was stopped due to an 'artificial' constraint, i.e. not
+// a condition imposed / required by the underlying algorithm, but instead an
+// engineering / implementation consideration. But a step which exceeds the
+// minimum step size, and satsifies the Armijo condition was still found,
+// and should thus be used [zoom not required].
+//
+// Returns false if no step size > minimum step size was found which
+// satisfies at least the Armijo condition.
+bool WolfeLineSearch::BracketingPhase(
+ const FunctionSample& initial_position,
+ const double step_size_estimate,
+ FunctionSample* bracket_low,
+ FunctionSample* bracket_high,
+ bool* do_zoom_search,
+ Summary* summary) const {
+ LineSearchFunction* function = options().function;
+
+ FunctionSample previous = initial_position;
+ FunctionSample current;
+
+ const double descent_direction_max_norm =
+ function->DirectionInfinityNorm();
+
+ *do_zoom_search = false;
+ *bracket_low = initial_position;
+
+ // As we require the gradient to evaluate the Wolfe condition, we always
+ // calculate it together with the value, irrespective of the interpolation
+ // type. As opposed to only calculating the gradient after the Armijo
+ // condition is satisifed, as the computational saving from this approach
+ // would be slight (perhaps even negative due to the extra call). Also,
+ // always calculating the value & gradient together protects against us
+ // reporting invalid solutions if the cost function returns slightly different
+ // function values when evaluated with / without gradients (due to numerical
+ // issues).
+ ++summary->num_function_evaluations;
+ ++summary->num_gradient_evaluations;
+ const bool kEvaluateGradient = true;
+ function->Evaluate(step_size_estimate, kEvaluateGradient, ¤t);
+ while (true) {
+ ++summary->num_iterations;
+
+ if (current.value_is_valid &&
+ (current.value > (initial_position.value
+ + options().sufficient_decrease
+ * initial_position.gradient
+ * current.x) ||
+ (previous.value_is_valid && current.value > previous.value))) {
+ // Bracket found: current step size violates Armijo sufficient decrease
+ // condition, or has stepped past an inflection point of f() relative to
+ // previous step size.
+ *do_zoom_search = true;
+ *bracket_low = previous;
+ *bracket_high = current;
+ VLOG(3) << std::scientific
+ << std::setprecision(kErrorMessageNumericPrecision)
+ << "Bracket found: current step (" << current.x
+ << ") violates Armijo sufficient condition, or has passed an "
+ << "inflection point of f() based on value.";
+ break;
+ }
+
+ if (current.value_is_valid &&
+ fabs(current.gradient) <=
+ -options().sufficient_curvature_decrease * initial_position.gradient) {
+ // Current step size satisfies the strong Wolfe conditions, and is thus a
+ // valid termination point, therefore a Zoom not required.
+ *bracket_low = current;
+ *bracket_high = current;
+ VLOG(3) << std::scientific
+ << std::setprecision(kErrorMessageNumericPrecision)
+ << "Bracketing phase found step size: " << current.x
+ << ", satisfying strong Wolfe conditions, initial_position: "
+ << initial_position << ", current: " << current;
+ break;
+
+ } else if (current.value_is_valid && current.gradient >= 0) {
+ // Bracket found: current step size has stepped past an inflection point
+ // of f(), but Armijo sufficient decrease is still satisfied and
+ // f(current) is our best minimum thus far. Remember step size
+ // monotonically increases, thus previous_step_size < current_step_size
+ // even though f(previous) > f(current).
+ *do_zoom_search = true;
+ // Note inverse ordering from first bracket case.
+ *bracket_low = current;
+ *bracket_high = previous;
+ VLOG(3) << "Bracket found: current step (" << current.x
+ << ") satisfies Armijo, but has gradient >= 0, thus have passed "
+ << "an inflection point of f().";
+ break;
+
+ } else if (current.value_is_valid &&
+ fabs(current.x - previous.x) * descent_direction_max_norm
+ < options().min_step_size) {
+ // We have shrunk the search bracket to a width less than our tolerance,
+ // and still not found either a point satisfying the strong Wolfe
+ // conditions, or a valid bracket containing such a point. Stop searching
+ // and set bracket_low to the size size amongst all those tested which
+ // minimizes f() and satisfies the Armijo condition.
+ LOG_IF(WARNING, !options().is_silent)
+ << "Line search failed: Wolfe bracketing phase shrank "
+ << "bracket width: " << fabs(current.x - previous.x)
+ << ", to < tolerance: " << options().min_step_size
+ << ", with descent_direction_max_norm: "
+ << descent_direction_max_norm << ", and failed to find "
+ << "a point satisfying the strong Wolfe conditions or a "
+ << "bracketing containing such a point. Accepting "
+ << "point found satisfying Armijo condition only, to "
+ << "allow continuation.";
+ *bracket_low = current;
+ break;
+
+ } else if (summary->num_iterations >= options().max_num_iterations) {
+ // Check num iterations bound here so that we always evaluate the
+ // max_num_iterations-th iteration against all conditions, and
+ // then perform no additional (unused) evaluations.
+ summary->error =
+ StringPrintf("Line search failed: Wolfe bracketing phase failed to "
+ "find a point satisfying strong Wolfe conditions, or a "
+ "bracket containing such a point within specified "
+ "max_num_iterations: %d", options().max_num_iterations);
+ LOG_IF(WARNING, !options().is_silent) << summary->error;
+ // Ensure that bracket_low is always set to the step size amongst all
+ // those tested which minimizes f() and satisfies the Armijo condition
+ // when we terminate due to the 'artificial' max_num_iterations condition.
+ *bracket_low =
+ current.value_is_valid && current.value < bracket_low->value
+ ? current : *bracket_low;
+ break;
+ }
+ // Either: f(current) is invalid; or, f(current) is valid, but does not
+ // satisfy the strong Wolfe conditions itself, or the conditions for
+ // being a boundary of a bracket.
+
+ // If f(current) is valid, (but meets no criteria) expand the search by
+ // increasing the step size. If f(current) is invalid, contract the step
+ // size.
+ //
+ // In Nocedal & Wright [1] (p60), the step-size can only increase in the
+ // bracketing phase: step_size_{k+1} \in [step_size_k, step_size_k * factor].
+ // However this does not account for the function returning invalid values
+ // which we support, in which case we need to contract the step size whilst
+ // ensuring that we do not invert the bracket, i.e, we require that:
+ // step_size_{k-1} <= step_size_{k+1} < step_size_k.
+ const double min_step_size =
+ current.value_is_valid
+ ? current.x : previous.x;
+ const double max_step_size =
+ current.value_is_valid
+ ? (current.x * options().max_step_expansion) : current.x;
+
+ // We are performing 2-point interpolation only here, but the API of
+ // InterpolatingPolynomialMinimizingStepSize() allows for up to
+ // 3-point interpolation, so pad call with a sample with an invalid
+ // value that will therefore be ignored.
+ const FunctionSample unused_previous;
+ DCHECK(!unused_previous.value_is_valid);
+ // Contracts step size if f(current) is not valid.
+ const double polynomial_minimization_start_time = WallTimeInSeconds();
+ const double step_size =
+ this->InterpolatingPolynomialMinimizingStepSize(
+ options().interpolation_type,
+ previous,
+ unused_previous,
+ current,
+ min_step_size,
+ max_step_size);
+ summary->polynomial_minimization_time_in_seconds +=
+ (WallTimeInSeconds() - polynomial_minimization_start_time);
+ if (step_size * descent_direction_max_norm < options().min_step_size) {
+ summary->error =
+ StringPrintf("Line search failed: step_size too small: %.5e "
+ "with descent_direction_max_norm: %.5e", step_size,
+ descent_direction_max_norm);
+ LOG_IF(WARNING, !options().is_silent) << summary->error;
+ return false;
+ }
+
+ // Only advance the lower boundary (in x) of the bracket if f(current)
+ // is valid such that we can support contracting the step size when
+ // f(current) is invalid without risking inverting the bracket in x, i.e.
+ // prevent previous.x > current.x.
+ previous = current.value_is_valid ? current : previous;
+ ++summary->num_function_evaluations;
+ ++summary->num_gradient_evaluations;
+ function->Evaluate(step_size, kEvaluateGradient, ¤t);
+ }
+
+ // Ensure that even if a valid bracket was found, we will only mark a zoom
+ // as required if the bracket's width is greater than our minimum tolerance.
+ if (*do_zoom_search &&
+ fabs(bracket_high->x - bracket_low->x) * descent_direction_max_norm
+ < options().min_step_size) {
+ *do_zoom_search = false;
+ }
+
+ return true;
+}
+
+// Returns true iff solution satisfies the strong Wolfe conditions. Otherwise,
+// on return false, if we stopped searching due to the 'artificial' condition of
+// reaching max_num_iterations, solution is the step size amongst all those
+// tested, which satisfied the Armijo decrease condition and minimized f().
+bool WolfeLineSearch::ZoomPhase(const FunctionSample& initial_position,
+ FunctionSample bracket_low,
+ FunctionSample bracket_high,
+ FunctionSample* solution,
+ Summary* summary) const {
+ LineSearchFunction* function = options().function;
+
+ CHECK(bracket_low.value_is_valid && bracket_low.gradient_is_valid)
+ << std::scientific << std::setprecision(kErrorMessageNumericPrecision)
+ << "Ceres bug: f_low input to Wolfe Zoom invalid, please contact "
+ << "the developers!, initial_position: " << initial_position
+ << ", bracket_low: " << bracket_low
+ << ", bracket_high: "<< bracket_high;
+ // We do not require bracket_high.gradient_is_valid as the gradient condition
+ // for a valid bracket is only dependent upon bracket_low.gradient, and
+ // in order to minimize jacobian evaluations, bracket_high.gradient may
+ // not have been calculated (if bracket_high.value does not satisfy the
+ // Armijo sufficient decrease condition and interpolation method does not
+ // require it).
+ //
+ // We also do not require that: bracket_low.value < bracket_high.value,
+ // although this is typical. This is to deal with the case when
+ // bracket_low = initial_position, bracket_high is the first sample,
+ // and bracket_high does not satisfy the Armijo condition, but still has
+ // bracket_high.value < initial_position.value.
+ CHECK(bracket_high.value_is_valid)
+ << std::scientific << std::setprecision(kErrorMessageNumericPrecision)
+ << "Ceres bug: f_high input to Wolfe Zoom invalid, please "
+ << "contact the developers!, initial_position: " << initial_position
+ << ", bracket_low: " << bracket_low
+ << ", bracket_high: "<< bracket_high;
+
+ if (bracket_low.gradient * (bracket_high.x - bracket_low.x) >= 0) {
+ // The third condition for a valid initial bracket:
+ //
+ // 3. bracket_high is chosen after bracket_low, s.t.
+ // bracket_low.gradient * (bracket_high.x - bracket_low.x) < 0.
+ //
+ // is not satisfied. As this can happen when the users' cost function
+ // returns inconsistent gradient values relative to the function values,
+ // we do not CHECK_LT(), but we do stop processing and return an invalid
+ // value.
+ summary->error =
+ StringPrintf("Line search failed: Wolfe zoom phase passed a bracket "
+ "which does not satisfy: bracket_low.gradient * "
+ "(bracket_high.x - bracket_low.x) < 0 [%.8e !< 0] "
+ "with initial_position: %s, bracket_low: %s, bracket_high:"
+ " %s, the most likely cause of which is the cost function "
+ "returning inconsistent gradient & function values.",
+ bracket_low.gradient * (bracket_high.x - bracket_low.x),
+ initial_position.ToDebugString().c_str(),
+ bracket_low.ToDebugString().c_str(),
+ bracket_high.ToDebugString().c_str());
+ LOG_IF(WARNING, !options().is_silent) << summary->error;
+ solution->value_is_valid = false;
+ return false;
+ }
+
+ const int num_bracketing_iterations = summary->num_iterations;
+ const double descent_direction_max_norm = function->DirectionInfinityNorm();
+
+ while (true) {
+ // Set solution to bracket_low, as it is our best step size (smallest f())
+ // found thus far and satisfies the Armijo condition, even though it does
+ // not satisfy the Wolfe condition.
+ *solution = bracket_low;
+ if (summary->num_iterations >= options().max_num_iterations) {
+ summary->error =
+ StringPrintf("Line search failed: Wolfe zoom phase failed to "
+ "find a point satisfying strong Wolfe conditions "
+ "within specified max_num_iterations: %d, "
+ "(num iterations taken for bracketing: %d).",
+ options().max_num_iterations, num_bracketing_iterations);
+ LOG_IF(WARNING, !options().is_silent) << summary->error;
+ return false;
+ }
+ if (fabs(bracket_high.x - bracket_low.x) * descent_direction_max_norm
+ < options().min_step_size) {
+ // Bracket width has been reduced below tolerance, and no point satisfying
+ // the strong Wolfe conditions has been found.
+ summary->error =
+ StringPrintf("Line search failed: Wolfe zoom bracket width: %.5e "
+ "too small with descent_direction_max_norm: %.5e.",
+ fabs(bracket_high.x - bracket_low.x),
+ descent_direction_max_norm);
+ LOG_IF(WARNING, !options().is_silent) << summary->error;
+ return false;
+ }
+
+ ++summary->num_iterations;
+ // Polynomial interpolation requires inputs ordered according to step size,
+ // not f(step size).
+ const FunctionSample& lower_bound_step =
+ bracket_low.x < bracket_high.x ? bracket_low : bracket_high;
+ const FunctionSample& upper_bound_step =
+ bracket_low.x < bracket_high.x ? bracket_high : bracket_low;
+ // We are performing 2-point interpolation only here, but the API of
+ // InterpolatingPolynomialMinimizingStepSize() allows for up to
+ // 3-point interpolation, so pad call with a sample with an invalid
+ // value that will therefore be ignored.
+ const FunctionSample unused_previous;
+ DCHECK(!unused_previous.value_is_valid);
+ const double polynomial_minimization_start_time = WallTimeInSeconds();
+ const double step_size =
+ this->InterpolatingPolynomialMinimizingStepSize(
+ options().interpolation_type,
+ lower_bound_step,
+ unused_previous,
+ upper_bound_step,
+ lower_bound_step.x,
+ upper_bound_step.x);
+ summary->polynomial_minimization_time_in_seconds +=
+ (WallTimeInSeconds() - polynomial_minimization_start_time);
+ // No check on magnitude of step size being too small here as it is
+ // lower-bounded by the initial bracket start point, which was valid.
+ //
+ // As we require the gradient to evaluate the Wolfe condition, we always
+ // calculate it together with the value, irrespective of the interpolation
+ // type. As opposed to only calculating the gradient after the Armijo
+ // condition is satisifed, as the computational saving from this approach
+ // would be slight (perhaps even negative due to the extra call). Also,
+ // always calculating the value & gradient together protects against us
+ // reporting invalid solutions if the cost function returns slightly
+ // different function values when evaluated with / without gradients (due
+ // to numerical issues).
+ ++summary->num_function_evaluations;
+ ++summary->num_gradient_evaluations;
+ const bool kEvaluateGradient = true;
+ function->Evaluate(step_size, kEvaluateGradient, solution);
+ if (!solution->value_is_valid || !solution->gradient_is_valid) {
+ summary->error =
+ StringPrintf("Line search failed: Wolfe Zoom phase found "
+ "step_size: %.5e, for which function is invalid, "
+ "between low_step: %.5e and high_step: %.5e "
+ "at which function is valid.",
+ solution->x, bracket_low.x, bracket_high.x);
+ LOG_IF(WARNING, !options().is_silent) << summary->error;
+ return false;
+ }
+
+ VLOG(3) << "Zoom iteration: "
+ << summary->num_iterations - num_bracketing_iterations
+ << ", bracket_low: " << bracket_low
+ << ", bracket_high: " << bracket_high
+ << ", minimizing solution: " << *solution;
+
+ if ((solution->value > (initial_position.value
+ + options().sufficient_decrease
+ * initial_position.gradient
+ * solution->x)) ||
+ (solution->value >= bracket_low.value)) {
+ // Armijo sufficient decrease not satisfied, or not better
+ // than current lowest sample, use as new upper bound.
+ bracket_high = *solution;
+ continue;
+ }
+
+ // Armijo sufficient decrease satisfied, check strong Wolfe condition.
+ if (fabs(solution->gradient) <=
+ -options().sufficient_curvature_decrease * initial_position.gradient) {
+ // Found a valid termination point satisfying strong Wolfe conditions.
+ VLOG(3) << std::scientific
+ << std::setprecision(kErrorMessageNumericPrecision)
+ << "Zoom phase found step size: " << solution->x
+ << ", satisfying strong Wolfe conditions.";
+ break;
+
+ } else if (solution->gradient * (bracket_high.x - bracket_low.x) >= 0) {
+ bracket_high = bracket_low;
+ }
+
+ bracket_low = *solution;
+ }
+ // Solution contains a valid point which satisfies the strong Wolfe
+ // conditions.
+ return true;
+}
+
+} // namespace internal
+} // namespace ceres