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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: sameeragarwal@google.com (Sameer Agarwal)
#include <list>
#include "ceres/internal/eigen.h"
#include "ceres/low_rank_inverse_hessian.h"
#include "glog/logging.h"
namespace ceres {
namespace internal {
using std::list;
// The (L)BFGS algorithm explicitly requires that the secant equation:
//
// B_{k+1} * s_k = y_k
//
// Is satisfied at each iteration, where B_{k+1} is the approximated
// Hessian at the k+1-th iteration, s_k = (x_{k+1} - x_{k}) and
// y_k = (grad_{k+1} - grad_{k}). As the approximated Hessian must be
// positive definite, this is equivalent to the condition:
//
// s_k^T * y_k > 0 [s_k^T * B_{k+1} * s_k = s_k^T * y_k > 0]
//
// This condition would always be satisfied if the function was strictly
// convex, alternatively, it is always satisfied provided that a Wolfe line
// search is used (even if the function is not strictly convex). See [1]
// (p138) for a proof.
//
// Although Ceres will always use a Wolfe line search when using (L)BFGS,
// practical implementation considerations mean that the line search
// may return a point that satisfies only the Armijo condition, and thus
// could violate the Secant equation. As such, we will only use a step
// to update the Hessian approximation if:
//
// s_k^T * y_k > tolerance
//
// It is important that tolerance is very small (and >=0), as otherwise we
// might skip the update too often and fail to capture important curvature
// information in the Hessian. For example going from 1e-10 -> 1e-14 improves
// the NIST benchmark score from 43/54 to 53/54.
//
// [1] Nocedal J., Wright S., Numerical Optimization, 2nd Ed. Springer, 1999.
//
// TODO(alexs.mac): Consider using Damped BFGS update instead of
// skipping update.
const double kLBFGSSecantConditionHessianUpdateTolerance = 1e-14;
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),
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) {
}
bool LowRankInverseHessian::Update(const Vector& delta_x,
const Vector& delta_gradient) {
const double delta_x_dot_delta_gradient = delta_x.dot(delta_gradient);
if (delta_x_dot_delta_gradient <=
kLBFGSSecantConditionHessianUpdateTolerance) {
VLOG(2) << "Skipping L-BFGS Update, delta_x_dot_delta_gradient too "
<< "small: " << delta_x_dot_delta_gradient << ", tolerance: "
<< kLBFGSSecantConditionHessianUpdateTolerance
<< " (Secant condition).";
return false;
}
int next = indices_.size();
// Once the size of the list reaches max_num_corrections_, simulate
// a circular buffer by removing the first element of the list and
// making it the next position where the LBFGS history is stored.
if (next == max_num_corrections_) {
next = indices_.front();
indices_.pop_front();
}
indices_.push_back(next);
delta_x_history_.col(next) = delta_x;
delta_gradient_history_.col(next) = delta_gradient;
delta_x_dot_delta_gradient_(next) = delta_x_dot_delta_gradient;
approximate_eigenvalue_scale_ =
delta_x_dot_delta_gradient / delta_gradient.squaredNorm();
return true;
}
void LowRankInverseHessian::RightMultiply(const double* x_ptr,
double* y_ptr) const {
ConstVectorRef gradient(x_ptr, num_parameters_);
VectorRef search_direction(y_ptr, num_parameters_);
search_direction = gradient;
const int num_corrections = indices_.size();
Vector alpha(num_corrections);
for (list<int>::const_reverse_iterator it = indices_.rbegin();
it != indices_.rend();
++it) {
const double alpha_i = delta_x_history_.col(*it).dot(search_direction) /
delta_x_dot_delta_gradient_(*it);
search_direction -= alpha_i * delta_gradient_history_.col(*it);
alpha(*it) = alpha_i;
}
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_;
VLOG(4) << "Applying approximate_eigenvalue_scale: "
<< approximate_eigenvalue_scale_ << " to initial inverse Hessian "
<< "approximation.";
}
for (list<int>::const_iterator it = indices_.begin();
it != indices_.end();
++it) {
const double beta = delta_gradient_history_.col(*it).dot(search_direction) /
delta_x_dot_delta_gradient_(*it);
search_direction += delta_x_history_.col(*it) * (alpha(*it) - beta);
}
}
} // namespace internal
} // namespace ceres
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