Adding Wolfe line search algorithm and full BFGS search direction options.
Change-Id: I9d3fb117805bdfa5bc33613368f45ae8f10e0d79
diff --git a/docs/source/bibliography.rst b/docs/source/bibliography.rst
index 30639d1..7ba435a 100644
--- a/docs/source/bibliography.rst
+++ b/docs/source/bibliography.rst
@@ -80,6 +80,10 @@
.. [NocedalWright] J. Nocedal & S. Wright, **Numerical Optimization**,
Springer, 2004.
+.. [Oren] S. S. Oren, **Self-scaling Variable Metric (SSVM) Algorithms
+ Part II: Implementation and Experiments**, Management Science,
+ 20(5), 863-874, 1974.
+
.. [RuheWedin] A. Ruhe and P.Å. Wedin, **Algorithms for separable
nonlinear least squares problems**, Siam Review, 22(3):318–337,
1980.
diff --git a/docs/source/solving.rst b/docs/source/solving.rst
index 99189e7..86a931a 100644
--- a/docs/source/solving.rst
+++ b/docs/source/solving.rst
@@ -386,12 +386,20 @@
``FLETCHER_REEVES``, ``POLAK_RIBIRERE`` and ``HESTENES_STIEFEL``
directions.
-3. ``LBFGS`` In this method, a limited memory approximation to the
- inverse Hessian is maintained and used to compute a quasi-Newton
- step [Nocedal]_, [ByrdNocedal]_.
+3. ``BFGS`` A generalization of the Secant method to multiple dimensions in
+ which a full, dense approximation to the inverse Hessian is maintained and
+ used to compute a quasi-Newton step [NocedalWright]_. BFGS is currently the best
+ known general quasi-Newton algorithm.
-Currently Ceres Solver uses a backtracking and interpolation based
-Armijo line search algorithm.
+4. ``LBFGS`` A limited memory approximation to the full ``BFGS`` method in
+ which the last `M` iterations are used to approximate the inverse Hessian
+ used to compute a quasi-Newton step [Nocedal]_, [ByrdNocedal]_.
+
+Currently Ceres Solver supports both a backtracking and interpolation
+based Armijo line search algorithm, and a sectioning / zoom interpolation
+(strong) Wolfe condition line search algorithm. However, note that in order for
+the assumptions underlying the ``BFGS`` and ``LBFGS`` methods to be
+guaranteed to be satisfied the Wolfe line search algorithm should be used.
.. _section-linear-solver:
@@ -780,14 +788,17 @@
Default: ``LBFGS``
- Choices are ``STEEPEST_DESCENT``, ``NONLINEAR_CONJUGATE_GRADIENT``
- and ``LBFGS``.
+ Choices are ``STEEPEST_DESCENT``, ``NONLINEAR_CONJUGATE_GRADIENT``,
+ ``BFGS`` and ``LBFGS``.
.. member:: LineSearchType Solver::Options::line_search_type
- Default: ``ARMIJO``
+ Default: ``WOLFE``
- ``ARMIJO`` is the only choice right now.
+ Choices are ``ARMIJO`` and ``WOLFE`` (strong Wolfe conditions). Note that in
+ order for the assumptions underlying the ``BFGS`` and ``LBFGS`` line search
+ direction algorithms to be guaranteed to be satisifed, the ``WOLFE`` line search
+ should be used.
.. member:: NonlinearConjugateGradientType Solver::Options::nonlinear_conjugate_gradient_type
@@ -800,7 +811,7 @@
Default: 20
- The LBFGS hessian approximation is a low rank approximation to the
+ The L-BFGS hessian approximation is a low rank approximation to the
inverse of the Hessian matrix. The rank of the approximation
determines (linearly) the space and time complexity of using the
approximation. Higher the rank, the better is the quality of the
@@ -819,6 +830,40 @@
rank. The best choice usually requires some problem specific
experimentation.
+.. member:: bool Solver::Options::use_approximate_eigenvalue_bfgs_scaling
+
+ Default: ``false``
+
+ As part of the ``BFGS`` update step / ``LBFGS`` right-multiply step,
+ the initial inverse Hessian approximation is taken to be the Identity.
+ However, [Oren]_ showed that using instead :math:`I * \gamma`, where
+ :math:`\gamma` is a scalar chosen to approximate an eigenvalue of the true
+ inverse Hessian can result in improved convergence in a wide variety of cases.
+ Setting ``use_approximate_eigenvalue_bfgs_scaling`` to true enables this
+ scaling in ``BFGS`` (before first iteration) and ``LBFGS`` (at each iteration).
+
+ Precisely, approximate eigenvalue scaling equates to
+
+ .. math:: \gamma = \frac{y_k' s_k}{y_k' y_k}
+
+ With:
+
+ .. math:: y_k = \nabla f_{k+1} - \nabla f_k
+ .. math:: s_k = x_{k+1} - x_k
+
+ Where :math:`f()` is the line search objective and :math:`x` the vector of
+ parameter values [NocedalWright]_.
+
+ It is important to note that approximate eigenvalue scaling does **not**
+ *always* improve convergence, and that it can in fact *significantly* degrade
+ performance for certain classes of problem, which is why it is disabled
+ by default. In particular it can degrade performance when the
+ sensitivity of the problem to different parameters varies significantly,
+ as in this case a single scalar factor fails to capture this variation
+ and detrimentally downscales parts of the Jacobian approximation which
+ correspond to low-sensitivity parameters. It can also reduce the
+ robustness of the solution to errors in the Jacobians.
+
.. member:: LineSearchIterpolationType Solver::Options::line_search_interpolation_type
Default: ``CUBIC``
@@ -829,10 +874,16 @@
.. member:: double Solver::Options::min_line_search_step_size
- If during the line search, the step size falls below this value, it
- is truncated to zero.
+ The line search terminates if:
-.. member:: double Solver::Options::armijo_sufficient_decrease
+ .. math:: \|\Delta x_k\|_\infty < \text{min_line_search_step_size}
+
+ where :math:`\|\cdot\|_\infty` refers to the max norm, and :math:`\Delta x_k` is
+ the step change in the parameter values at the :math:`k`-th iteration.
+
+.. member:: double Solver::Options::line_search_sufficient_function_decrease
+
+ Default: ``1e-4``
Solving the line search problem exactly is computationally
prohibitive. Fortunately, line search based optimization algorithms
@@ -843,17 +894,90 @@
.. math:: f(\text{step_size}) \le f(0) + \text{sufficient_decrease} * [f'(0) * \text{step_size}]
-.. member:: double Solver::Options::min_armijo_relative_step_size_change
+ This condition is known as the Armijo condition.
- In each iteration of the Armijo line search,
+.. member:: double Solver::Options::max_line_search_step_contraction
- .. math:: \text{new_step_size} \ge \text{min_relative_step_size_change} * \text{step_size}
+ Default: ``1e-3``
-.. member:: double Solver::Options::max_armijo_relative_step_size_change
+ In each iteration of the line search,
- In each iteration of the Armijo line search,
+ .. math:: \text{new_step_size} >= \text{max_line_search_step_contraction} * \text{step_size}
- .. math:: \text{new_step_size} \le \text{max_relative_step_size_change} * \text{step_size}
+ Note that by definition, for contraction:
+
+ .. math:: 0 < \text{max_step_contraction} < \text{min_step_contraction} < 1
+
+.. member:: double Solver::Options::min_line_search_step_contraction
+
+ Default: ``0.6``
+
+ In each iteration of the line search,
+
+ .. math:: \text{new_step_size} <= \text{min_line_search_step_contraction} * \text{step_size}
+
+ Note that by definition, for contraction:
+
+ .. math:: 0 < \text{max_step_contraction} < \text{min_step_contraction} < 1
+
+.. member:: int Solver::Options::max_num_line_search_step_size_iterations
+
+ Default: ``20``
+
+ 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 stop.
+
+ As this is an 'artificial' constraint (one imposed by the user, not the underlying math),
+ if ``WOLFE`` line search is being used, *and* points satisfying the Armijo sufficient
+ (function) decrease condition have been found during the current search
+ (in :math:`<=` ``max_num_line_search_step_size_iterations``). Then, the step
+ size with the lowest function value which satisfies the Armijo condition will be
+ returned as the new valid step, even though it does *not* satisfy the strong Wolfe
+ conditions. This behaviour protects against early termination of the optimizer at a
+ sub-optimal point.
+
+.. member:: int Solver::Options::max_num_line_search_direction_restarts
+
+ Default: ``5``
+
+ Maximum number of restarts of the line search direction algorithm before
+ terminating the optimization. Restarts of the line search direction
+ algorithm occur when the current algorithm fails to produce a new descent
+ direction. This typically indicates a numerical failure, or a breakdown
+ in the validity of the approximations used.
+
+.. member:: double Solver::Options::line_search_sufficient_curvature_decrease
+
+ Default: ``0.9``
+
+ 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.
+
+ .. math:: \|f'(\text{step_size})\| <= \text{sufficient_curvature_decrease} * \|f'(0)\|
+
+ Where :math:`f()` is the line search objective and :math:`f'()` is the derivative
+ of :math:`f` with respect to the step size: :math:`\frac{d f}{d~\text{step size}}`.
+
+.. member:: double Solver::Options::max_line_search_step_expansion
+
+ Default: ``10.0``
+
+ During the bracketing phase of a Wolfe line 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:
+
+ .. math:: \text{new_step_size} <= \text{max_step_expansion} * \text{step_size}
+
+ By definition for expansion
+
+ .. math:: \text{max_step_expansion} > 1.0
.. member:: TrustRegionStrategyType Solver::Options::trust_region_strategy_type
diff --git a/examples/CMakeLists.txt b/examples/CMakeLists.txt
index e9e5cef..9dfc80b 100644
--- a/examples/CMakeLists.txt
+++ b/examples/CMakeLists.txt
@@ -37,9 +37,6 @@
ADD_EXECUTABLE(helloworld_analytic_diff helloworld_analytic_diff.cc)
TARGET_LINK_LIBRARIES(helloworld_analytic_diff ceres)
-ADD_EXECUTABLE(powell powell.cc)
-TARGET_LINK_LIBRARIES(powell ceres)
-
ADD_EXECUTABLE(curve_fitting curve_fitting.cc)
TARGET_LINK_LIBRARIES(curve_fitting ceres)
@@ -55,6 +52,9 @@
TARGET_LINK_LIBRARIES(simple_bundle_adjuster ceres)
IF (GFLAGS)
+ ADD_EXECUTABLE(powell powell.cc)
+ TARGET_LINK_LIBRARIES(powell ceres)
+
ADD_EXECUTABLE(nist nist.cc)
TARGET_LINK_LIBRARIES(nist ceres)
diff --git a/examples/nist.cc b/examples/nist.cc
index 8e0f37e..1773a0f 100644
--- a/examples/nist.cc
+++ b/examples/nist.cc
@@ -81,6 +81,8 @@
DEFINE_string(nist_data_dir, "", "Directory containing the NIST non-linear"
"regression examples");
+DEFINE_string(minimizer, "trust_region",
+ "Minimizer type to use, choices are: line_search & trust_region");
DEFINE_string(trust_region_strategy, "levenberg_marquardt",
"Options are: levenberg_marquardt, dogleg");
DEFINE_string(dogleg, "traditional_dogleg",
@@ -90,6 +92,25 @@
"cgnr");
DEFINE_string(preconditioner, "jacobi", "Options are: "
"identity, jacobi");
+DEFINE_string(line_search, "armijo",
+ "Line search algorithm to use, choices are: armijo and wolfe.");
+DEFINE_string(line_search_direction, "lbfgs",
+ "Line search direction algorithm to use, choices: lbfgs, bfgs");
+DEFINE_int32(max_line_search_iterations, 20,
+ "Maximum number of iterations for each line search.");
+DEFINE_int32(max_line_search_restarts, 10,
+ "Maximum number of restarts of line search direction algorithm.");
+DEFINE_string(line_search_interpolation, "cubic",
+ "Degree of polynomial aproximation in line search, "
+ "choices are: bisection, quadratic & cubic.");
+DEFINE_int32(lbfgs_rank, 20,
+ "Rank of L-BFGS inverse Hessian approximation in line search.");
+DEFINE_bool(approximate_eigenvalue_bfgs_scaling, false,
+ "Use approximate eigenvalue scaling in (L)BFGS line search.");
+DEFINE_double(sufficient_decrease, 1.0e-4,
+ "Line search Armijo sufficient (function) decrease factor.");
+DEFINE_double(sufficient_curvature_decrease, 0.9,
+ "Line search Wolfe sufficient curvature decrease factor.");
DEFINE_int32(num_iterations, 10000, "Number of iterations");
DEFINE_bool(nonmonotonic_steps, false, "Trust region algorithm can use"
" nonmonotic steps");
@@ -392,7 +413,7 @@
template <typename Model, int num_residuals, int num_parameters>
int RegressionDriver(const std::string& filename,
- const ceres::Solver::Options& options) {
+ const ceres::Solver::Options& options) {
NISTProblem nist_problem(FLAGS_nist_data_dir + filename);
CHECK_EQ(num_residuals, nist_problem.response_size());
CHECK_EQ(num_parameters, nist_problem.num_parameters());
@@ -446,18 +467,22 @@
++num_success;
}
- printf("start: %d status: %s lre: %4.1f initial cost: %e final cost:%e certified cost: %e\n",
+ printf("start: %d status: %s lre: %4.1f initial cost: %e final cost:%e "
+ "certified cost: %e total iterations: %d\n",
start + 1,
log_relative_error < kMinNumMatchingDigits ? "FAILURE" : "SUCCESS",
log_relative_error,
summary.initial_cost,
summary.final_cost,
- nist_problem.certified_cost());
+ nist_problem.certified_cost(),
+ (summary.num_successful_steps + summary.num_unsuccessful_steps));
}
return num_success;
}
void SetMinimizerOptions(ceres::Solver::Options* options) {
+ CHECK(ceres::StringToMinimizerType(FLAGS_minimizer,
+ &options->minimizer_type));
CHECK(ceres::StringToLinearSolverType(FLAGS_linear_solver,
&options->linear_solver_type));
CHECK(ceres::StringToPreconditionerType(FLAGS_preconditioner,
@@ -466,10 +491,28 @@
FLAGS_trust_region_strategy,
&options->trust_region_strategy_type));
CHECK(ceres::StringToDoglegType(FLAGS_dogleg, &options->dogleg_type));
+ CHECK(ceres::StringToLineSearchDirectionType(
+ FLAGS_line_search_direction,
+ &options->line_search_direction_type));
+ CHECK(ceres::StringToLineSearchType(FLAGS_line_search,
+ &options->line_search_type));
+ CHECK(ceres::StringToLineSearchInterpolationType(
+ FLAGS_line_search_interpolation,
+ &options->line_search_interpolation_type));
options->max_num_iterations = FLAGS_num_iterations;
options->use_nonmonotonic_steps = FLAGS_nonmonotonic_steps;
options->initial_trust_region_radius = FLAGS_initial_trust_region_radius;
+ options->max_lbfgs_rank = FLAGS_lbfgs_rank;
+ options->line_search_sufficient_function_decrease = FLAGS_sufficient_decrease;
+ options->line_search_sufficient_curvature_decrease =
+ FLAGS_sufficient_curvature_decrease;
+ options->max_num_line_search_step_size_iterations =
+ FLAGS_max_line_search_iterations;
+ options->max_num_line_search_direction_restarts =
+ FLAGS_max_line_search_restarts;
+ options->use_approximate_eigenvalue_bfgs_scaling =
+ FLAGS_approximate_eigenvalue_bfgs_scaling;
options->function_tolerance = 1e-18;
options->gradient_tolerance = 1e-18;
options->parameter_tolerance = 1e-18;
diff --git a/examples/powell.cc b/examples/powell.cc
index 4a41728..c0cba02 100644
--- a/examples/powell.cc
+++ b/examples/powell.cc
@@ -46,6 +46,7 @@
#include <vector>
#include "ceres/ceres.h"
+#include "gflags/gflags.h"
#include "glog/logging.h"
using ceres::AutoDiffCostFunction;
@@ -94,7 +95,11 @@
}
};
+DEFINE_string(minimizer, "trust_region",
+ "Minimizer type to use, choices are: line_search & trust_region");
+
int main(int argc, char** argv) {
+ google::ParseCommandLineFlags(&argc, &argv, true);
google::InitGoogleLogging(argv[0]);
double x1 = 3.0;
@@ -119,23 +124,27 @@
NULL,
&x1, &x4);
- // Run the solver!
Solver::Options options;
- options.max_num_iterations = 30;
+ LOG_IF(FATAL, !ceres::StringToMinimizerType(FLAGS_minimizer,
+ &options.minimizer_type))
+ << "Invalid minimizer: " << FLAGS_minimizer
+ << ", valid options are: trust_region and line_search.";
+
+ options.max_num_iterations = 100;
options.linear_solver_type = ceres::DENSE_QR;
options.minimizer_progress_to_stdout = true;
- Solver::Summary summary;
-
std::cout << "Initial x1 = " << x1
<< ", x2 = " << x2
<< ", x3 = " << x3
<< ", x4 = " << x4
<< "\n";
+ // Run the solver!
+ Solver::Summary summary;
Solve(options, &problem, &summary);
- std::cout << summary.BriefReport() << "\n";
+ std::cout << summary.FullReport() << "\n";
std::cout << "Final x1 = " << x1
<< ", x2 = " << x2
<< ", x3 = " << x3
diff --git a/include/ceres/iteration_callback.h b/include/ceres/iteration_callback.h
index 13140c2..987c2d9 100644
--- a/include/ceres/iteration_callback.h
+++ b/include/ceres/iteration_callback.h
@@ -54,6 +54,8 @@
eta(0.0),
step_size(0.0),
line_search_function_evaluations(0),
+ line_search_gradient_evaluations(0),
+ line_search_iterations(0),
linear_solver_iterations(0),
iteration_time_in_seconds(0.0),
step_solver_time_in_seconds(0.0),
@@ -121,9 +123,15 @@
// Step sized computed by the line search algorithm.
double step_size;
- // Number of function evaluations used by the line search algorithm.
+ // Number of function value evaluations used by the line search algorithm.
int line_search_function_evaluations;
+ // Number of function gradient evaluations used by the line search algorithm.
+ int line_search_gradient_evaluations;
+
+ // Number of iterations taken by the line search algorithm.
+ int line_search_iterations;
+
// Number of iterations taken by the linear solver to solve for the
// Newton step.
int linear_solver_iterations;
diff --git a/include/ceres/solver.h b/include/ceres/solver.h
index 47c7e94..e7b8d09 100644
--- a/include/ceres/solver.h
+++ b/include/ceres/solver.h
@@ -60,14 +60,19 @@
Options() {
minimizer_type = TRUST_REGION;
line_search_direction_type = LBFGS;
- line_search_type = ARMIJO;
+ line_search_type = WOLFE;
nonlinear_conjugate_gradient_type = FLETCHER_REEVES;
max_lbfgs_rank = 20;
+ use_approximate_eigenvalue_bfgs_scaling = false;
line_search_interpolation_type = CUBIC;
min_line_search_step_size = 1e-9;
- armijo_sufficient_decrease = 1e-4;
- min_armijo_relative_step_size_change = 1e-3;
- max_armijo_relative_step_size_change = 0.6;
+ line_search_sufficient_function_decrease = 1e-4;
+ max_line_search_step_contraction = 1e-3;
+ min_line_search_step_contraction = 0.6;
+ max_num_line_search_step_size_iterations = 20;
+ max_num_line_search_direction_restarts = 5;
+ line_search_sufficient_curvature_decrease = 0.9;
+ max_line_search_step_expansion = 10.0;
trust_region_strategy_type = LEVENBERG_MARQUARDT;
dogleg_type = TRADITIONAL_DOGLEG;
@@ -178,6 +183,28 @@
// Limited Storage". Mathematics of Computation 35 (151): 773–782.
int max_lbfgs_rank;
+ // As part of the (L)BFGS update step (BFGS) / right-multiply step (L-BFGS),
+ // the initial inverse Hessian approximation is taken to be the Identity.
+ // However, Oren showed that using instead I * \gamma, where \gamma is
+ // chosen to approximate an eigenvalue of the true inverse Hessian can
+ // result in improved convergence in a wide variety of cases. Setting
+ // use_approximate_eigenvalue_bfgs_scaling to true enables this scaling.
+ //
+ // It is important to note that approximate eigenvalue scaling does not
+ // always improve convergence, and that it can in fact significantly degrade
+ // performance for certain classes of problem, which is why it is disabled
+ // by default. In particular it can degrade performance when the
+ // sensitivity of the problem to different parameters varies significantly,
+ // as in this case a single scalar factor fails to capture this variation
+ // and detrimentally downscales parts of the jacobian approximation which
+ // correspond to low-sensitivity parameters. It can also reduce the
+ // robustness of the solution to errors in the jacobians.
+ //
+ // Oren S.S., Self-scaling variable metric (SSVM) algorithms
+ // Part II: Implementation and experiments, Management Science,
+ // 20(5), 863-874, 1974.
+ bool use_approximate_eigenvalue_bfgs_scaling;
+
// Degree of the polynomial used to approximate the objective
// function. Valid values are BISECTION, QUADRATIC and CUBIC.
//
@@ -189,7 +216,7 @@
// value, it is truncated to zero.
double min_line_search_step_size;
- // Armijo line search parameters.
+ // Line search parameters.
// Solving the line search problem exactly is computationally
// prohibitive. Fortunately, line search based optimization
@@ -201,19 +228,63 @@
//
// f(step_size) <= f(0) + sufficient_decrease * f'(0) * step_size
//
- double armijo_sufficient_decrease;
+ double line_search_sufficient_function_decrease;
- // In each iteration of the Armijo line search,
+ // In each iteration of the line search,
//
- // new_step_size >= min_relative_step_size_change * step_size
+ // new_step_size >= max_line_search_step_contraction * step_size
//
- double min_armijo_relative_step_size_change;
+ // Note that by definition, for contraction:
+ //
+ // 0 < max_step_contraction < min_step_contraction < 1
+ //
+ double max_line_search_step_contraction;
- // In each iteration of the Armijo line search,
+ // In each iteration of the line search,
//
- // new_step_size <= max_relative_step_size_change * step_size
+ // new_step_size <= min_line_search_step_contraction * step_size
//
- double max_armijo_relative_step_size_change;
+ // Note that by definition, for contraction:
+ //
+ // 0 < max_step_contraction < min_step_contraction < 1
+ //
+ double min_line_search_step_contraction;
+
+ // 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_line_search_step_size_iterations;
+
+ // Maximum number of restarts of the line search direction algorithm before
+ // terminating the optimization. Restarts of the line search direction
+ // algorithm occur when the current algorithm fails to produce a new descent
+ // direction. This typically indicates a numerical failure, or a breakdown
+ // in the validity of the approximations used.
+ int max_num_line_search_direction_restarts;
+
+ // 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 line_search_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_line_search_step_expansion;
TrustRegionStrategyType trust_region_strategy_type;
diff --git a/include/ceres/types.h b/include/ceres/types.h
index 5becb53..46c2fd2 100644
--- a/include/ceres/types.h
+++ b/include/ceres/types.h
@@ -37,6 +37,8 @@
#ifndef CERES_PUBLIC_TYPES_H_
#define CERES_PUBLIC_TYPES_H_
+#include <string>
+
#include "ceres/internal/port.h"
namespace ceres {
@@ -167,10 +169,47 @@
// used is determined by NonlinerConjuateGradientType.
NONLINEAR_CONJUGATE_GRADIENT,
- // A limited memory approximation to the inverse Hessian is
- // maintained and used to compute a quasi-Newton step.
+ // BFGS, and it's limited memory approximation L-BFGS, are quasi-Newton
+ // algorithms that approximate the Hessian matrix by iteratively refining
+ // an initial estimate with rank-one updates using the gradient at each
+ // iteration. They are a generalisation of the Secant method and satisfy
+ // the Secant equation. The Secant equation has an infinium of solutions
+ // in multiple dimensions, as there are N*(N+1)/2 degrees of freedom in a
+ // symmetric matrix but only N conditions are specified by the Secant
+ // equation. The requirement that the Hessian approximation be positive
+ // definite imposes another N additional constraints, but that still leaves
+ // remaining degrees-of-freedom. (L)BFGS methods uniquely deteremine the
+ // approximate Hessian by imposing the additional constraints that the
+ // approximation at the next iteration must be the 'closest' to the current
+ // approximation (the nature of how this proximity is measured is actually
+ // the defining difference between a family of quasi-Newton methods including
+ // (L)BFGS & DFP). (L)BFGS is currently regarded as being the best known
+ // general quasi-Newton method.
//
- // For more details see
+ // The principal difference between BFGS and L-BFGS is that whilst BFGS
+ // maintains a full, dense approximation to the (inverse) Hessian, L-BFGS
+ // maintains only a window of the last M observations of the parameters and
+ // gradients. Using this observation history, the calculation of the next
+ // search direction can be computed without requiring the construction of the
+ // full dense inverse Hessian approximation. This is particularly important
+ // for problems with a large number of parameters, where storage of an N-by-N
+ // matrix in memory would be prohibitive.
+ //
+ // For more details on BFGS see:
+ //
+ // Broyden, C.G., "The Convergence of a Class of Double-rank Minimization
+ // Algorithms,"; J. Inst. Maths. Applics., Vol. 6, pp 76–90, 1970.
+ //
+ // Fletcher, R., "A New Approach to Variable Metric Algorithms,"
+ // Computer Journal, Vol. 13, pp 317–322, 1970.
+ //
+ // Goldfarb, D., "A Family of Variable Metric Updates Derived by Variational
+ // Means," Mathematics of Computing, Vol. 24, pp 23–26, 1970.
+ //
+ // Shanno, D.F., "Conditioning of Quasi-Newton Methods for Function
+ // Minimization," Mathematics of Computing, Vol. 24, pp 647–656, 1970.
+ //
+ // For more details on L-BFGS see:
//
// Nocedal, J. (1980). "Updating Quasi-Newton Matrices with Limited
// Storage". Mathematics of Computation 35 (151): 773–782.
@@ -179,7 +218,12 @@
// "Representations of Quasi-Newton Matrices and their use in
// Limited Memory Methods". Mathematical Programming 63 (4):
// 129–156.
+ //
+ // A general reference for both methods:
+ //
+ // Nocedal J., Wright S., Numerical Optimization, 2nd Ed. Springer, 1999.
LBFGS,
+ BFGS,
};
// Nonliner conjugate gradient methods are a generalization of the
@@ -198,6 +242,7 @@
// Backtracking line search with polynomial interpolation or
// bisection.
ARMIJO,
+ WOLFE,
};
// Ceres supports different strategies for computing the trust region
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;
}
