Adding Wolfe line search algorithm and full BFGS search direction options.
Change-Id: I9d3fb117805bdfa5bc33613368f45ae8f10e0d79
diff --git a/internal/ceres/line_search.cc b/internal/ceres/line_search.cc
index 06c823f..2c75d89 100644
--- a/internal/ceres/line_search.cc
+++ b/internal/ceres/line_search.cc
@@ -35,6 +35,7 @@
#include "ceres/evaluator.h"
#include "ceres/internal/eigen.h"
#include "ceres/polynomial.h"
+#include "ceres/stringprintf.h"
#include "glog/logging.h"
namespace ceres {
@@ -61,8 +62,41 @@
return sample;
};
+// Convenience stream operator for pushing FunctionSamples into log messages.
+std::ostream& operator<<(std::ostream &os,
+ const FunctionSample& sample) {
+ os << "[x: " << sample.x << ", value: " << sample.value
+ << ", gradient: " << sample.gradient << ", value_is_valid: "
+ << std::boolalpha << sample.value_is_valid << ", gradient_is_valid: "
+ << std::boolalpha << sample.gradient_is_valid << "]";
+ return os;
+};
+
} // namespace
+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()),
@@ -103,104 +137,608 @@
return IsFinite(*f) && IsFinite(*g);
}
-void ArmijoLineSearch::Search(const LineSearch::Options& options,
- const double initial_step_size,
+double LineSearchFunction::DirectionInfinityNorm() const {
+ return direction_.lpNorm<Eigen::Infinity>();
+}
+
+// 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 min(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)
+ << "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(ValueSample(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(ValueSample(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::Search(const double step_size_estimate,
const double initial_cost,
const double initial_gradient,
Summary* summary) {
*CHECK_NOTNULL(summary) = LineSearch::Summary();
- Function* function = options.function;
+ 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);
+ Function* function = options().function;
- double previous_step_size = 0.0;
- double previous_cost = 0.0;
- double previous_gradient = 0.0;
- bool previous_step_size_is_valid = false;
+ // Note initial_cost & initial_gradient are evaluated at step_size = 0,
+ // not step_size_estimate, which is our starting guess.
+ const FunctionSample initial_position =
+ ValueAndGradientSample(0.0, initial_cost, initial_gradient);
- double step_size = initial_step_size;
- double cost = 0.0;
- double gradient = 0.0;
- bool step_size_is_valid = false;
+ FunctionSample previous = ValueAndGradientSample(0.0, 0.0, 0.0);
+ previous.value_is_valid = false;
- ++summary->num_evaluations;
- step_size_is_valid =
- function->Evaluate(step_size,
- &cost,
- options.interpolation_type != CUBIC ? NULL : &gradient);
- while (!step_size_is_valid || cost > (initial_cost
- + options.sufficient_decrease
- * initial_gradient
- * step_size)) {
- // If step_size_is_valid is not true we treat it as if the cost at
- // that point is not large enough to satisfy the sufficient
- // decrease condition.
+ FunctionSample current = ValueAndGradientSample(step_size_estimate, 0.0, 0.0);
+ current.value_is_valid = false;
- const double current_step_size = step_size;
- // Backtracking search. Each iteration of this loop finds a new point
+ const bool interpolation_uses_gradients =
+ options().interpolation_type == CUBIC;
+ const double descent_direction_max_norm =
+ static_cast<const LineSearchFunction*>(function)->DirectionInfinityNorm();
- if ((options.interpolation_type == BISECTION) || !step_size_is_valid) {
- step_size *= 0.5;
- } else {
- // Backtrack by interpolating the function and gradient values
- // and minimizing the corresponding polynomial.
- vector<FunctionSample> samples;
- samples.push_back(ValueAndGradientSample(0.0,
- initial_cost,
- initial_gradient));
-
- if (options.interpolation_type == QUADRATIC) {
- // Two point interpolation using function values and the
- // initial gradient.
- samples.push_back(ValueSample(step_size, cost));
-
- if (summary->num_evaluations > 1 && previous_step_size_is_valid) {
- // Three point interpolation, using function values and the
- // initial gradient.
- samples.push_back(ValueSample(previous_step_size, previous_cost));
- }
- } else {
- // Two point interpolation using the function values and the gradients.
- samples.push_back(ValueAndGradientSample(step_size,
- cost,
- gradient));
-
- if (summary->num_evaluations > 1 && previous_step_size_is_valid) {
- // Three point interpolation using the function values and
- // the gradients.
- samples.push_back(ValueAndGradientSample(previous_step_size,
- previous_cost,
- previous_gradient));
- }
- }
-
- double min_value;
- MinimizeInterpolatingPolynomial(samples, 0.0, current_step_size,
- &step_size, &min_value);
- step_size =
- min(max(step_size,
- options.min_relative_step_size_change * current_step_size),
- options.max_relative_step_size_change * current_step_size);
- }
-
- previous_step_size = current_step_size;
- previous_cost = cost;
- previous_gradient = gradient;
-
- if (fabs(initial_gradient) * step_size < options.min_step_size) {
- LOG(WARNING) << "Line search failed: step_size too small: " << step_size;
+ ++summary->num_function_evaluations;
+ if (interpolation_uses_gradients) { ++summary->num_gradient_evaluations; }
+ current.value_is_valid =
+ function->Evaluate(current.x,
+ ¤t.value,
+ interpolation_uses_gradients
+ ? ¤t.gradient : NULL);
+ current.gradient_is_valid =
+ interpolation_uses_gradients && current.value_is_valid;
+ 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(WARNING) << summary->error;
return;
}
- ++summary->num_evaluations;
- step_size_is_valid =
- function->Evaluate(step_size,
- &cost,
- options.interpolation_type != CUBIC ? NULL : &gradient);
+ 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));
+
+ 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(WARNING) << summary->error;
+ return;
+ }
+
+ previous = current;
+ current.x = step_size;
+
+ ++summary->num_function_evaluations;
+ if (interpolation_uses_gradients) { ++summary->num_gradient_evaluations; }
+ current.value_is_valid =
+ function->Evaluate(current.x,
+ ¤t.value,
+ interpolation_uses_gradients
+ ? ¤t.gradient : NULL);
+ current.gradient_is_valid =
+ interpolation_uses_gradients && current.value_is_valid;
}
- summary->optimal_step_size = step_size;
+ summary->optimal_step_size = current.x;
summary->success = true;
}
+WolfeLineSearch::WolfeLineSearch(const LineSearch::Options& options)
+ : LineSearch(options) {}
+
+void WolfeLineSearch::Search(const double step_size_estimate,
+ const double initial_cost,
+ const double initial_gradient,
+ Summary* summary) {
+ *CHECK_NOTNULL(summary) = LineSearch::Summary();
+ // 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.
+ const FunctionSample initial_position =
+ ValueAndGradientSample(0.0, initial_cost, initial_gradient);
+
+ 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) &&
+ summary->num_iterations < options().max_num_iterations) {
+ // Failed to find either a valid point or a valid bracket, but we did not
+ // run out of iterations.
+ 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. As this is an
+ // 'artificial' constraint, and we would otherwise fail to produce a valid
+ // point when ArmijoLineSearch would succeed, we return the lowest point
+ // found thus far which satsifies the Armijo condition (but not the Wolfe
+ // conditions).
+ CHECK(bracket_low.value_is_valid)
+ << "Ceres bug: Bracketing produced an invalid bracket_low, please "
+ << "contact the developers!, bracket_low: " << bracket_low
+ << ", bracket_high: " << bracket_high << ", num_iterations: "
+ << summary->num_iterations << ", max_num_iterations: "
+ << options().max_num_iterations;
+ summary->optimal_step_size = bracket_low.x;
+ summary->success = true;
+ return;
+ }
+
+ // 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.
+ solution =
+ solution.value_is_valid && solution.value <= bracket_low.value
+ ? solution : bracket_low;
+
+ summary->optimal_step_size = solution.x;
+ summary->success = true;
+}
+
+// Returns true iff bracket_low & bracket_high bound a bracket that contains
+// points which satisfy the strong Wolfe conditions. Otherwise, on return false,
+// if we stopped searching due to the 'artificial' condition of reaching
+// max_num_iterations, bracket_low is the step size amongst all those
+// tested, which satisfied the Armijo decrease condition and minimized f().
+bool WolfeLineSearch::BracketingPhase(
+ const FunctionSample& initial_position,
+ const double step_size_estimate,
+ FunctionSample* bracket_low,
+ FunctionSample* bracket_high,
+ bool* do_zoom_search,
+ Summary* summary) {
+ Function* function = options().function;
+
+ FunctionSample previous = initial_position;
+ FunctionSample current = ValueAndGradientSample(step_size_estimate, 0.0, 0.0);
+ current.value_is_valid = false;
+
+ const bool interpolation_uses_gradients =
+ options().interpolation_type == CUBIC;
+ const double descent_direction_max_norm =
+ static_cast<const LineSearchFunction*>(function)->DirectionInfinityNorm();
+
+ *do_zoom_search = false;
+ *bracket_low = initial_position;
+
+ ++summary->num_function_evaluations;
+ if (interpolation_uses_gradients) { ++summary->num_gradient_evaluations; }
+ current.value_is_valid =
+ function->Evaluate(current.x,
+ ¤t.value,
+ interpolation_uses_gradients
+ ? ¤t.gradient : NULL);
+ current.gradient_is_valid =
+ interpolation_uses_gradients && current.value_is_valid;
+
+ 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;
+ break;
+ }
+
+ // Irrespective of the interpolation type we are using, we now need the
+ // gradient at the current point (which satisfies the Armijo condition)
+ // in order to check the strong Wolfe conditions.
+ if (!interpolation_uses_gradients) {
+ ++summary->num_function_evaluations;
+ ++summary->num_gradient_evaluations;
+ current.value_is_valid =
+ function->Evaluate(current.x,
+ ¤t.value,
+ ¤t.gradient);
+ current.gradient_is_valid = current.value_is_valid;
+ }
+
+ 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;
+ 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;
+ 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(WARNING) << 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;
+ return false;
+ }
+ // 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.
+ 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 step_size =
+ this->InterpolatingPolynomialMinimizingStepSize(
+ options().interpolation_type,
+ previous,
+ unused_previous,
+ current,
+ previous.x,
+ max_step_size);
+ 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(WARNING) << summary->error;
+ return false;
+ }
+
+ previous = current.value_is_valid ? current : previous;
+ current.x = step_size;
+
+ ++summary->num_function_evaluations;
+ if (interpolation_uses_gradients) { ++summary->num_gradient_evaluations; }
+ current.value_is_valid =
+ function->Evaluate(current.x,
+ ¤t.value,
+ interpolation_uses_gradients
+ ? ¤t.gradient : NULL);
+ current.gradient_is_valid =
+ interpolation_uses_gradients && current.value_is_valid;
+ }
+ // Either we have a valid point, defined as a bracket of zero width, in which
+ // case no zoom is required, or a valid bracket in which to zoom.
+ 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) {
+ Function* function = options().function;
+
+ CHECK(bracket_low.value_is_valid && bracket_low.gradient_is_valid)
+ << "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).
+ CHECK(bracket_high.value_is_valid)
+ << "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;
+ CHECK_LT(bracket_low.gradient *
+ (bracket_high.x - bracket_low.x), 0.0)
+ << "Ceres bug: f_high input to Wolfe Zoom does not satisfy gradient "
+ << "condition combined with f_low, please contact the developers!"
+ << ", initial_position: " << initial_position
+ << ", bracket_low: " << bracket_low
+ << ", bracket_high: "<< bracket_high;
+
+ const int num_bracketing_iterations = summary->num_iterations;
+ const bool interpolation_uses_gradients =
+ options().interpolation_type == CUBIC;
+ const double descent_direction_max_norm =
+ static_cast<const LineSearchFunction*>(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(WARNING) << 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(WARNING) << 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);
+ solution->x =
+ this->InterpolatingPolynomialMinimizingStepSize(
+ options().interpolation_type,
+ lower_bound_step,
+ unused_previous,
+ upper_bound_step,
+ lower_bound_step.x,
+ upper_bound_step.x);
+ // 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.
+ ++summary->num_function_evaluations;
+ if (interpolation_uses_gradients) { ++summary->num_gradient_evaluations; }
+ solution->value_is_valid =
+ function->Evaluate(solution->x,
+ &solution->value,
+ interpolation_uses_gradients
+ ? &solution->gradient : NULL);
+ solution->gradient_is_valid =
+ interpolation_uses_gradients && solution->value_is_valid;
+ if (!solution->value_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(WARNING) << summary->error;
+ return false;
+ }
+
+ 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 (!interpolation_uses_gradients) {
+ // Irrespective of the interpolation type we are using, we now need the
+ // gradient at the current point (which satisfies the Armijo condition)
+ // in order to check the strong Wolfe conditions.
+ ++summary->num_function_evaluations;
+ ++summary->num_gradient_evaluations;
+ solution->value_is_valid =
+ function->Evaluate(solution->x,
+ &solution->value,
+ &solution->gradient);
+ solution->gradient_is_valid = solution->value_is_valid;
+ if (!solution->value_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(WARNING) << summary->error;
+ return false;
+ }
+ }
+ if (fabs(solution->gradient) <=
+ -options().sufficient_curvature_decrease * initial_position.gradient) {
+ // Found a valid termination point 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
diff --git a/internal/ceres/line_search.h b/internal/ceres/line_search.h
index 5792652..e4836b2 100644
--- a/internal/ceres/line_search.h
+++ b/internal/ceres/line_search.h
@@ -35,6 +35,7 @@
#ifndef CERES_NO_LINE_SEARCH_MINIMIZER
+#include <string>
#include <vector>
#include "ceres/internal/eigen.h"
#include "ceres/internal/port.h"
@@ -44,6 +45,7 @@
namespace internal {
class Evaluator;
+struct FunctionSample;
// Line search is another name for a one dimensional optimization
// algorithm. The name "line search" comes from the fact one
@@ -63,16 +65,19 @@
Options()
: interpolation_type(CUBIC),
sufficient_decrease(1e-4),
- min_relative_step_size_change(1e-3),
- max_relative_step_size_change(0.9),
+ max_step_contraction(1e-3),
+ min_step_contraction(0.9),
min_step_size(1e-9),
+ max_num_iterations(20),
+ sufficient_curvature_decrease(0.9),
+ max_step_expansion(10.0),
function(NULL) {}
// Degree of the polynomial used to approximate the objective
// function.
LineSearchInterpolationType interpolation_type;
- // Armijo line search parameters.
+ // Armijo and Wolfe line search parameters.
// Solving the line search problem exactly is computationally
// prohibitive. Fortunately, line search based optimization
@@ -85,20 +90,60 @@
// f(step_size) <= f(0) + sufficient_decrease * f'(0) * step_size
double sufficient_decrease;
- // In each iteration of the Armijo line search,
+ // In each iteration of the Armijo / Wolfe line search,
//
- // new_step_size >= min_relative_step_size_change * step_size
- double min_relative_step_size_change;
+ // 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;
- // In each iteration of the Armijo line search,
+ // In each iteration of the Armijo / Wolfe line search,
//
- // new_step_size <= max_relative_step_size_change * step_size
- double max_relative_step_size_change;
+ // 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;
// If during the line search, the step_size falls below this
// value, it is truncated to zero.
double min_step_size;
+ // 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;
+
+ // 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;
+
+ // 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;
+
// The one dimensional function that the line search algorithm
// minimizes.
Function* function;
@@ -133,18 +178,28 @@
Summary()
: success(false),
optimal_step_size(0.0),
- num_evaluations(0) {}
+ num_function_evaluations(0),
+ num_gradient_evaluations(0),
+ num_iterations(0) {}
bool success;
double optimal_step_size;
- int num_evaluations;
+ int num_function_evaluations;
+ int num_gradient_evaluations;
+ int num_iterations;
+ string error;
};
+ explicit LineSearch(const LineSearch::Options& options);
virtual ~LineSearch() {}
+ static LineSearch* Create(const LineSearchType line_search_type,
+ const LineSearch::Options& options,
+ string* error);
+
// Perform the line search.
//
- // initial_step_size must be a positive number.
+ // step_size_estimate must be a positive number.
//
// initial_cost and initial_gradient are the values and gradient of
// the function at zero.
@@ -152,11 +207,23 @@
// search.
//
// Summary::success is true if a non-zero step size is found.
- virtual void Search(const LineSearch::Options& options,
- double initial_step_size,
+ virtual void Search(double step_size_estimate,
double initial_cost,
double initial_gradient,
Summary* summary) = 0;
+ 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:
+ LineSearch::Options options_;
};
class LineSearchFunction : public LineSearch::Function {
@@ -165,6 +232,7 @@
virtual ~LineSearchFunction() {}
void Init(const Vector& position, const Vector& direction);
virtual bool Evaluate(const double x, double* f, double* g);
+ double DirectionInfinityNorm() const;
private:
Evaluator* evaluator_;
@@ -186,14 +254,44 @@
// 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() {}
- virtual void Search(const LineSearch::Options& options,
- double initial_step_size,
+ virtual void Search(double step_size_estimate,
double initial_cost,
double initial_gradient,
Summary* summary);
};
+// 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() {}
+ virtual void Search(double step_size_estimate,
+ double initial_cost,
+ double initial_gradient,
+ Summary* summary);
+ // 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);
+ // 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);
+};
+
} // namespace internal
} // namespace ceres
diff --git a/internal/ceres/line_search_direction.cc b/internal/ceres/line_search_direction.cc
index b8b582c..8ded823 100644
--- a/internal/ceres/line_search_direction.cc
+++ b/internal/ceres/line_search_direction.cc
@@ -100,14 +100,24 @@
class LBFGS : public LineSearchDirection {
public:
- LBFGS(const int num_parameters, const int max_lbfgs_rank)
- : low_rank_inverse_hessian_(num_parameters, max_lbfgs_rank) {}
+ LBFGS(const int num_parameters,
+ const int max_lbfgs_rank,
+ const bool use_approximate_eigenvalue_bfgs_scaling)
+ : low_rank_inverse_hessian_(num_parameters,
+ max_lbfgs_rank,
+ use_approximate_eigenvalue_bfgs_scaling),
+ is_positive_definite_(true) {}
virtual ~LBFGS() {}
bool NextDirection(const LineSearchMinimizer::State& previous,
const LineSearchMinimizer::State& current,
Vector* search_direction) {
+ CHECK(is_positive_definite_)
+ << "Ceres bug: NextDirection() called on L-BFGS after inverse Hessian "
+ << "approximation has become indefinite, please contact the "
+ << "developers!";
+
low_rank_inverse_hessian_.Update(
previous.search_direction * previous.step_size,
current.gradient - previous.gradient);
@@ -115,11 +125,177 @@
low_rank_inverse_hessian_.RightMultiply(current.gradient.data(),
search_direction->data());
*search_direction *= -1.0;
+
+ if (search_direction->dot(current.gradient) >= 0.0) {
+ LOG(WARNING) << "Numerical failure in L-BFGS update: inverse Hessian "
+ << "approximation is not positive definite, and thus "
+ << "initial gradient for search direction is positive: "
+ << search_direction->dot(current.gradient);
+ is_positive_definite_ = false;
+ return false;
+ }
+
return true;
}
private:
LowRankInverseHessian low_rank_inverse_hessian_;
+ bool is_positive_definite_;
+};
+
+class BFGS : public LineSearchDirection {
+ public:
+ BFGS(const int num_parameters,
+ const bool use_approximate_eigenvalue_scaling)
+ : num_parameters_(num_parameters),
+ use_approximate_eigenvalue_scaling_(use_approximate_eigenvalue_scaling),
+ initialized_(false),
+ is_positive_definite_(true) {
+ LOG_IF(WARNING, num_parameters_ >= 1e3)
+ << "BFGS line search being created with: " << num_parameters_
+ << " parameters, this will allocate a dense approximate inverse Hessian"
+ << " of size: " << num_parameters_ << " x " << num_parameters_
+ << ", consider using the L-BFGS memory-efficient line search direction "
+ << "instead.";
+ // Construct inverse_hessian_ after logging warning about size s.t. if the
+ // allocation crashes us, the log will highlight what the issue likely was.
+ inverse_hessian_ = Matrix::Identity(num_parameters, num_parameters);
+ }
+
+ virtual ~BFGS() {}
+
+ bool NextDirection(const LineSearchMinimizer::State& previous,
+ const LineSearchMinimizer::State& current,
+ Vector* search_direction) {
+ CHECK(is_positive_definite_)
+ << "Ceres bug: NextDirection() called on BFGS after inverse Hessian "
+ << "approximation has become indefinite, please contact the "
+ << "developers!";
+
+ const Vector delta_x = previous.search_direction * previous.step_size;
+ const Vector delta_gradient = current.gradient - previous.gradient;
+ const double delta_x_dot_delta_gradient = delta_x.dot(delta_gradient);
+
+ if (delta_x_dot_delta_gradient <= 1e-10) {
+ VLOG(2) << "Skipping BFGS Update, delta_x_dot_delta_gradient too "
+ << "small: " << delta_x_dot_delta_gradient;
+ } else {
+ // Update dense inverse Hessian approximation.
+
+ if (!initialized_ && use_approximate_eigenvalue_scaling_) {
+ // Rescale the initial inverse Hessian approximation (H_0) to be
+ // iteratively updated so that it is of similar 'size' to the true
+ // inverse Hessian at the start point. As shown in [1]:
+ //
+ // \gamma = (delta_gradient_{0}' * delta_x_{0}) /
+ // (delta_gradient_{0}' * delta_gradient_{0})
+ //
+ // Satisfies:
+ //
+ // (1 / \lambda_m) <= \gamma <= (1 / \lambda_1)
+ //
+ // Where \lambda_1 & \lambda_m are the smallest and largest eigenvalues
+ // of the true initial Hessian (not the inverse) respectively. Thus,
+ // \gamma is an approximate eigenvalue of the true inverse Hessian, and
+ // choosing: H_0 = I * \gamma will yield a starting point that has a
+ // similar scale to the true inverse Hessian. This technique is widely
+ // reported to often improve convergence, however this is not
+ // universally true, particularly if there are errors in the initial
+ // gradients, or if there are significant differences in the sensitivity
+ // of the problem to the parameters (i.e. the range of the magnitudes of
+ // the components of the gradient is large).
+ //
+ // The original origin of this rescaling trick is somewhat unclear, the
+ // earliest reference appears to be Oren [1], however it is widely
+ // discussed without specific attributation in various texts including
+ // [2] (p143).
+ //
+ // [1] Oren S.S., Self-scaling variable metric (SSVM) algorithms
+ // Part II: Implementation and experiments, Management Science,
+ // 20(5), 863-874, 1974.
+ // [2] Nocedal J., Wright S., Numerical Optimization, Springer, 1999.
+ inverse_hessian_ *=
+ delta_x_dot_delta_gradient / delta_gradient.dot(delta_gradient);
+ }
+ initialized_ = true;
+
+ // Efficient O(num_parameters^2) BFGS update [2].
+ //
+ // Starting from dense BFGS update detailed in Nocedal [2] p140/177 and
+ // using: y_k = delta_gradient, s_k = delta_x:
+ //
+ // \rho_k = 1.0 / (s_k' * y_k)
+ // V_k = I - \rho_k * y_k * s_k'
+ // H_k = (V_k' * H_{k-1} * V_k) + (\rho_k * s_k * s_k')
+ //
+ // This update involves matrix, matrix products which naively O(N^3),
+ // however we can exploit our knowledge that H_k is positive definite
+ // and thus by defn. symmetric to reduce the cost of the update:
+ //
+ // Expanding the update above yields:
+ //
+ // H_k = H_{k-1} +
+ // \rho_k * ( (1.0 + \rho_k * y_k' * H_k * y_k) * s_k * s_k' -
+ // (s_k * y_k' * H_k + H_k * y_k * s_k') )
+ //
+ // Using: A = (s_k * y_k' * H_k), and the knowledge that H_k = H_k', the
+ // last term simplifies to (A + A'). Note that although A is not symmetric
+ // (A + A') is symmetric. For ease of construction we also define
+ // B = (1 + \rho_k * y_k' * H_k * y_k) * s_k * s_k', which is by defn
+ // symmetric due to construction from: s_k * s_k'.
+ //
+ // Now we can write the BFGS update as:
+ //
+ // H_k = H_{k-1} + \rho_k * (B - (A + A'))
+
+ // For efficiency, as H_k is by defn. symmetric, we will only maintain the
+ // *lower* triangle of H_k (and all intermediary terms).
+
+ const double rho_k = 1.0 / delta_x_dot_delta_gradient;
+
+ // Calculate: A = s_k * y_k' * H_k
+ Matrix A = delta_x * (delta_gradient.transpose() *
+ inverse_hessian_.selfadjointView<Eigen::Lower>());
+
+ // Calculate scalar: (1 + \rho_k * y_k' * H_k * y_k)
+ const double delta_x_times_delta_x_transpose_scale_factor =
+ (1.0 + (rho_k * delta_gradient.transpose() *
+ inverse_hessian_.selfadjointView<Eigen::Lower>() *
+ delta_gradient));
+ // Calculate: B = (1 + \rho_k * y_k' * H_k * y_k) * s_k * s_k'
+ Matrix B = Matrix::Zero(num_parameters_, num_parameters_);
+ B.selfadjointView<Eigen::Lower>().
+ rankUpdate(delta_x, delta_x_times_delta_x_transpose_scale_factor);
+
+ // Finally, update inverse Hessian approximation according to:
+ // H_k = H_{k-1} + \rho_k * (B - (A + A')). Note that (A + A') is
+ // symmetric, even though A is not.
+ inverse_hessian_.triangularView<Eigen::Lower>() +=
+ rho_k * (B - A - A.transpose());
+ }
+
+ *search_direction =
+ inverse_hessian_.selfadjointView<Eigen::Lower>() *
+ (-1.0 * current.gradient);
+
+ if (search_direction->dot(current.gradient) >= 0.0) {
+ LOG(WARNING) << "Numerical failure in BFGS update: inverse Hessian "
+ << "approximation is not positive definite, and thus "
+ << "initial gradient for search direction is positive: "
+ << search_direction->dot(current.gradient);
+ is_positive_definite_ = false;
+ return false;
+ }
+
+ return true;
+ }
+
+ private:
+ const int num_parameters_;
+ const bool use_approximate_eigenvalue_scaling_;
+ Matrix inverse_hessian_;
+ bool initialized_;
+ bool is_positive_definite_;
};
LineSearchDirection*
@@ -135,8 +311,16 @@
}
if (options.type == ceres::LBFGS) {
- return new ceres::internal::LBFGS(options.num_parameters,
- options.max_lbfgs_rank);
+ return new ceres::internal::LBFGS(
+ options.num_parameters,
+ options.max_lbfgs_rank,
+ options.use_approximate_eigenvalue_bfgs_scaling);
+ }
+
+ if (options.type == ceres::BFGS) {
+ return new ceres::internal::BFGS(
+ options.num_parameters,
+ options.use_approximate_eigenvalue_bfgs_scaling);
}
LOG(ERROR) << "Unknown line search direction type: " << options.type;
diff --git a/internal/ceres/line_search_direction.h b/internal/ceres/line_search_direction.h
index 0874754..0857cb0 100644
--- a/internal/ceres/line_search_direction.h
+++ b/internal/ceres/line_search_direction.h
@@ -48,7 +48,8 @@
type(LBFGS),
nonlinear_conjugate_gradient_type(FLETCHER_REEVES),
function_tolerance(1e-12),
- max_lbfgs_rank(20) {
+ max_lbfgs_rank(20),
+ use_approximate_eigenvalue_bfgs_scaling(true) {
}
int num_parameters;
@@ -56,6 +57,7 @@
NonlinearConjugateGradientType nonlinear_conjugate_gradient_type;
double function_tolerance;
int max_lbfgs_rank;
+ bool use_approximate_eigenvalue_bfgs_scaling;
};
static LineSearchDirection* Create(const Options& options);
diff --git a/internal/ceres/line_search_minimizer.cc b/internal/ceres/line_search_minimizer.cc
index 24aada3..2cc89fa 100644
--- a/internal/ceres/line_search_minimizer.cc
+++ b/internal/ceres/line_search_minimizer.cc
@@ -160,6 +160,8 @@
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));
@@ -170,15 +172,32 @@
options.line_search_interpolation_type;
line_search_options.min_step_size = options.min_line_search_step_size;
line_search_options.sufficient_decrease =
- options.armijo_sufficient_decrease;
- line_search_options.min_relative_step_size_change =
- options.min_armijo_relative_step_size_change;
- line_search_options.max_relative_step_size_change =
- options.max_armijo_relative_step_size_change;
+ 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.function = &line_search_function;
- ArmijoLineSearch line_search;
+ scoped_ptr<LineSearch>
+ line_search(LineSearch::Create(options.line_search_type,
+ line_search_options,
+ &summary->error));
+ if (line_search.get() == NULL) {
+ LOG(ERROR) << "Ceres bug: Unable to create a LineSearch object, please "
+ << "contact the developers!, error: " << summary->error;
+ summary->termination_type = DID_NOT_RUN;
+ return;
+ }
+
LineSearch::Summary line_search_summary;
+ int num_line_search_direction_restarts = 0;
while (true) {
if (!RunCallbacks(options.callbacks, iteration_summary, summary)) {
@@ -215,9 +234,36 @@
¤t_state.search_direction);
}
- if (!line_search_status) {
- LOG(WARNING) << "Line search direction computation failed. "
- "Resorting to steepest descent.";
+ 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->error =
+ StringPrintf("Line search direction failure: specified "
+ "max_num_line_search_direction_restarts: %d reached.",
+ options.max_num_line_search_direction_restarts);
+ LOG(WARNING) << summary->error << " terminating optimization.";
+ summary->termination_type = NUMERICAL_FAILURE;
+ 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(WARNING)
+ << "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;
}
@@ -227,16 +273,34 @@
// TODO(sameeragarwal): Refactor this into its own object and add
// explanations for the various choices.
- const double initial_step_size = (iteration_summary.iteration == 1)
+ //
+ // 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)
? min(1.0, 1.0 / current_state.gradient_max_norm)
: 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->error =
+ 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));
+ LOG(WARNING) << summary->error;
+ summary->termination_type = NUMERICAL_FAILURE;
+ break;
+ }
- line_search.Search(line_search_options,
- initial_step_size,
- current_state.cost,
- current_state.directional_derivative,
- &line_search_summary);
+ line_search->Search(initial_step_size,
+ current_state.cost,
+ current_state.directional_derivative,
+ &line_search_summary);
current_state.step_size = line_search_summary.optimal_step_size;
delta = current_state.step_size * current_state.search_direction;
@@ -282,7 +346,11 @@
iteration_summary.step_norm = delta.norm();
iteration_summary.step_size = current_state.step_size;
iteration_summary.line_search_function_evaluations =
- line_search_summary.num_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 =
diff --git a/internal/ceres/low_rank_inverse_hessian.cc b/internal/ceres/low_rank_inverse_hessian.cc
index 4fabd5b..372165f 100644
--- a/internal/ceres/low_rank_inverse_hessian.cc
+++ b/internal/ceres/low_rank_inverse_hessian.cc
@@ -35,12 +35,15 @@
namespace ceres {
namespace internal {
-LowRankInverseHessian::LowRankInverseHessian(int num_parameters,
- int max_num_corrections)
+LowRankInverseHessian::LowRankInverseHessian(
+ int num_parameters,
+ int max_num_corrections,
+ bool use_approximate_eigenvalue_scaling)
: num_parameters_(num_parameters),
max_num_corrections_(max_num_corrections),
+ use_approximate_eigenvalue_scaling_(use_approximate_eigenvalue_scaling),
num_corrections_(0),
- diagonal_(1.0),
+ approximate_eigenvalue_scale_(1.0),
delta_x_history_(num_parameters, max_num_corrections),
delta_gradient_history_(num_parameters, max_num_corrections),
delta_x_dot_delta_gradient_(max_num_corrections) {
@@ -50,7 +53,8 @@
const Vector& delta_gradient) {
const double delta_x_dot_delta_gradient = delta_x.dot(delta_gradient);
if (delta_x_dot_delta_gradient <= 1e-10) {
- VLOG(2) << "Skipping LBFGS Update. " << delta_x_dot_delta_gradient;
+ VLOG(2) << "Skipping LBFGS Update, delta_x_dot_delta_gradient too small: "
+ << delta_x_dot_delta_gradient;
return false;
}
@@ -77,7 +81,8 @@
delta_gradient_history_.col(num_corrections_ - 1) = delta_gradient;
delta_x_dot_delta_gradient_(num_corrections_ - 1) =
delta_x_dot_delta_gradient;
- diagonal_ = delta_x_dot_delta_gradient / delta_gradient.squaredNorm();
+ approximate_eigenvalue_scale_ =
+ delta_x_dot_delta_gradient / delta_gradient.squaredNorm();
return true;
}
@@ -96,7 +101,39 @@
search_direction -= alpha(i) * delta_gradient_history_.col(i);
}
- search_direction *= diagonal_;
+ if (use_approximate_eigenvalue_scaling_) {
+ // Rescale the initial inverse Hessian approximation (H_0) to be iteratively
+ // updated so that it is of similar 'size' to the true inverse Hessian along
+ // the most recent search direction. As shown in [1]:
+ //
+ // \gamma_k = (delta_gradient_{k-1}' * delta_x_{k-1}) /
+ // (delta_gradient_{k-1}' * delta_gradient_{k-1})
+ //
+ // Satisfies:
+ //
+ // (1 / \lambda_m) <= \gamma_k <= (1 / \lambda_1)
+ //
+ // Where \lambda_1 & \lambda_m are the smallest and largest eigenvalues of
+ // the true Hessian (not the inverse) along the most recent search direction
+ // respectively. Thus \gamma is an approximate eigenvalue of the true
+ // inverse Hessian, and choosing: H_0 = I * \gamma will yield a starting
+ // point that has a similar scale to the true inverse Hessian. This
+ // technique is widely reported to often improve convergence, however this
+ // is not universally true, particularly if there are errors in the initial
+ // jacobians, or if there are significant differences in the sensitivity
+ // of the problem to the parameters (i.e. the range of the magnitudes of
+ // the components of the gradient is large).
+ //
+ // The original origin of this rescaling trick is somewhat unclear, the
+ // earliest reference appears to be Oren [1], however it is widely discussed
+ // without specific attributation in various texts including [2] (p143/178).
+ //
+ // [1] Oren S.S., Self-scaling variable metric (SSVM) algorithms Part II:
+ // Implementation and experiments, Management Science,
+ // 20(5), 863-874, 1974.
+ // [2] Nocedal J., Wright S., Numerical Optimization, Springer, 1999.
+ search_direction *= approximate_eigenvalue_scale_;
+ }
for (int i = 0; i < num_corrections_; ++i) {
const double beta = delta_gradient_history_.col(i).dot(search_direction) /
diff --git a/internal/ceres/low_rank_inverse_hessian.h b/internal/ceres/low_rank_inverse_hessian.h
index 6f3fc0c..7d293d0 100644
--- a/internal/ceres/low_rank_inverse_hessian.h
+++ b/internal/ceres/low_rank_inverse_hessian.h
@@ -61,10 +61,16 @@
public:
// num_parameters is the row/column size of the Hessian.
// max_num_corrections is the rank of the Hessian approximation.
+ // use_approximate_eigenvalue_scaling controls whether the initial
+ // inverse Hessian used during Right/LeftMultiply() is scaled by
+ // the approximate eigenvalue of the true inverse Hessian at the
+ // current operating point.
// The approximation uses:
// 2 * max_num_corrections * num_parameters + max_num_corrections
// doubles.
- LowRankInverseHessian(int num_parameters, int max_num_corrections);
+ LowRankInverseHessian(int num_parameters,
+ int max_num_corrections,
+ bool use_approximate_eigenvalue_scaling);
virtual ~LowRankInverseHessian() {}
// Update the low rank approximation. delta_x is the change in the
@@ -86,8 +92,9 @@
private:
const int num_parameters_;
const int max_num_corrections_;
+ const bool use_approximate_eigenvalue_scaling_;
int num_corrections_;
- double diagonal_;
+ double approximate_eigenvalue_scale_;
Matrix delta_x_history_;
Matrix delta_gradient_history_;
Vector delta_x_dot_delta_gradient_;
diff --git a/internal/ceres/minimizer.h b/internal/ceres/minimizer.h
index 300d2df..622e9ce 100644
--- a/internal/ceres/minimizer.h
+++ b/internal/ceres/minimizer.h
@@ -88,14 +88,25 @@
nonlinear_conjugate_gradient_type =
options.nonlinear_conjugate_gradient_type;
max_lbfgs_rank = options.max_lbfgs_rank;
+ use_approximate_eigenvalue_bfgs_scaling =
+ options.use_approximate_eigenvalue_bfgs_scaling;
line_search_interpolation_type =
options.line_search_interpolation_type;
min_line_search_step_size = options.min_line_search_step_size;
- armijo_sufficient_decrease = options.armijo_sufficient_decrease;
- min_armijo_relative_step_size_change =
- options.min_armijo_relative_step_size_change;
- max_armijo_relative_step_size_change =
- options.max_armijo_relative_step_size_change;
+ line_search_sufficient_function_decrease =
+ options.line_search_sufficient_function_decrease;
+ max_line_search_step_contraction =
+ options.max_line_search_step_contraction;
+ min_line_search_step_contraction =
+ options.min_line_search_step_contraction;
+ max_num_line_search_step_size_iterations =
+ options.max_num_line_search_step_size_iterations;
+ max_num_line_search_direction_restarts =
+ options.max_num_line_search_direction_restarts;
+ line_search_sufficient_curvature_decrease =
+ options.line_search_sufficient_curvature_decrease;
+ max_line_search_step_expansion =
+ options.max_line_search_step_expansion;
evaluator = NULL;
trust_region_strategy = NULL;
jacobian = NULL;
@@ -131,11 +142,16 @@
LineSearchType line_search_type;
NonlinearConjugateGradientType nonlinear_conjugate_gradient_type;
int max_lbfgs_rank;
+ bool use_approximate_eigenvalue_bfgs_scaling;
LineSearchInterpolationType line_search_interpolation_type;
double min_line_search_step_size;
- double armijo_sufficient_decrease;
- double min_armijo_relative_step_size_change;
- double max_armijo_relative_step_size_change;
+ double line_search_sufficient_function_decrease;
+ double max_line_search_step_contraction;
+ double min_line_search_step_contraction;
+ int max_num_line_search_step_size_iterations;
+ int max_num_line_search_direction_restarts;
+ double line_search_sufficient_curvature_decrease;
+ double max_line_search_step_expansion;
// List of callbacks that are executed by the Minimizer at the end
diff --git a/internal/ceres/solver_impl.cc b/internal/ceres/solver_impl.cc
index 3d58de7..8fc1305 100644
--- a/internal/ceres/solver_impl.cc
+++ b/internal/ceres/solver_impl.cc
@@ -636,6 +636,87 @@
summary->num_effective_parameters =
original_program->NumEffectiveParameters();
+ // Validate values for configuration parameters supplied by user.
+ if ((original_options.line_search_direction_type == ceres::BFGS ||
+ original_options.line_search_direction_type == ceres::LBFGS) &&
+ original_options.line_search_type != ceres::WOLFE) {
+ summary->error =
+ string("Invalid configuration: require line_search_type == "
+ "ceres::WOLFE when using (L)BFGS to ensure that underlying "
+ "assumptions are guaranteed to be satisfied.");
+ LOG(ERROR) << summary->error;
+ return;
+ }
+ if (original_options.max_lbfgs_rank == 0) {
+ summary->error =
+ string("Invalid configuration: require max_lbfgs_rank > 0");
+ LOG(ERROR) << summary->error;
+ return;
+ }
+ if (original_options.min_line_search_step_size <= 0.0) {
+ summary->error = "Invalid configuration: min_line_search_step_size <= 0.0.";
+ LOG(ERROR) << summary->error;
+ return;
+ }
+ if (original_options.line_search_sufficient_function_decrease <= 0.0) {
+ summary->error =
+ string("Invalid configuration: require ") +
+ string("line_search_sufficient_function_decrease <= 0.0.");
+ LOG(ERROR) << summary->error;
+ return;
+ }
+ if (original_options.max_line_search_step_contraction <= 0.0 ||
+ original_options.max_line_search_step_contraction >= 1.0) {
+ summary->error = string("Invalid configuration: require ") +
+ string("0.0 < max_line_search_step_contraction < 1.0.");
+ LOG(ERROR) << summary->error;
+ return;
+ }
+ if (original_options.min_line_search_step_contraction <=
+ original_options.max_line_search_step_contraction ||
+ original_options.min_line_search_step_contraction > 1.0) {
+ summary->error = string("Invalid configuration: require ") +
+ string("max_line_search_step_contraction < ") +
+ string("min_line_search_step_contraction <= 1.0.");
+ LOG(ERROR) << summary->error;
+ return;
+ }
+ // Warn user if they have requested BISECTION interpolation, but constraints
+ // on max/min step size change during line search prevent bisection scaling
+ // from occurring. Warn only, as this is likely a user mistake, but one which
+ // does not prevent us from continuing.
+ LOG_IF(WARNING,
+ (original_options.line_search_interpolation_type == ceres::BISECTION &&
+ (original_options.max_line_search_step_contraction > 0.5 ||
+ original_options.min_line_search_step_contraction < 0.5)))
+ << "Line search interpolation type is BISECTION, but specified "
+ << "max_line_search_step_contraction: "
+ << original_options.max_line_search_step_contraction << ", and "
+ << "min_line_search_step_contraction: "
+ << original_options.min_line_search_step_contraction
+ << ", prevent bisection (0.5) scaling, continuing with solve regardless.";
+ if (original_options.max_num_line_search_step_size_iterations <= 0) {
+ summary->error = string("Invalid configuration: require ") +
+ string("max_num_line_search_step_size_iterations > 0.");
+ LOG(ERROR) << summary->error;
+ return;
+ }
+ if (original_options.line_search_sufficient_curvature_decrease <=
+ original_options.line_search_sufficient_function_decrease ||
+ original_options.line_search_sufficient_curvature_decrease > 1.0) {
+ summary->error = string("Invalid configuration: require ") +
+ string("line_search_sufficient_function_decrease < ") +
+ string("line_search_sufficient_curvature_decrease < 1.0.");
+ LOG(ERROR) << summary->error;
+ return;
+ }
+ if (original_options.max_line_search_step_expansion <= 1.0) {
+ summary->error = string("Invalid configuration: require ") +
+ string("max_line_search_step_expansion > 1.0.");
+ LOG(ERROR) << summary->error;
+ return;
+ }
+
// Empty programs are usually a user error.
if (summary->num_parameter_blocks == 0) {
summary->error = "Problem contains no parameter blocks.";
diff --git a/internal/ceres/types.cc b/internal/ceres/types.cc
index e88cdd6..42990e3 100644
--- a/internal/ceres/types.cc
+++ b/internal/ceres/types.cc
@@ -165,6 +165,7 @@
CASESTR(STEEPEST_DESCENT);
CASESTR(NONLINEAR_CONJUGATE_GRADIENT);
CASESTR(LBFGS);
+ CASESTR(BFGS);
default:
return "UNKNOWN";
}
@@ -176,12 +177,14 @@
STRENUM(STEEPEST_DESCENT);
STRENUM(NONLINEAR_CONJUGATE_GRADIENT);
STRENUM(LBFGS);
+ STRENUM(BFGS);
return false;
}
const char* LineSearchTypeToString(LineSearchType type) {
switch (type) {
CASESTR(ARMIJO);
+ CASESTR(WOLFE);
default:
return "UNKNOWN";
}
@@ -190,6 +193,7 @@
bool StringToLineSearchType(string value, LineSearchType* type) {
UpperCase(&value);
STRENUM(ARMIJO);
+ STRENUM(WOLFE);
return false;
}
