Adding Wolfe line search algorithm and full BFGS search direction options.
Change-Id: I9d3fb117805bdfa5bc33613368f45ae8f10e0d79
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;
