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-// Copyright (C) 2008 Davis E. King (davis@dlib.net)
-// License: Boost Software License See LICENSE.txt for the full license.
-#ifndef DLIB_OPTIMIZATIOn_LINE_SEARCH_H_
-#define DLIB_OPTIMIZATIOn_LINE_SEARCH_H_
-
-#include <cmath>
-#include <limits>
-#include "../matrix.h"
-#include "../algs.h"
-#include "optimization_line_search_abstract.h"
-#include <utility>
-
-namespace dlib
-{
-
-// ----------------------------------------------------------------------------------------
-
- template <typename funct, typename T>
- class line_search_funct
- {
- public:
- line_search_funct(const funct& f_, const T& start_, const T& direction_)
- : f(f_),start(start_), direction(direction_), matrix_r(0), scalar_r(0)
- {}
-
- line_search_funct(const funct& f_, const T& start_, const T& direction_, T& r)
- : f(f_),start(start_), direction(direction_), matrix_r(&r), scalar_r(0)
- {}
-
- line_search_funct(const funct& f_, const T& start_, const T& direction_, double& r)
- : f(f_),start(start_), direction(direction_), matrix_r(0), scalar_r(&r)
- {}
-
- double operator()(const double& x) const
- {
- return get_value(f(start + x*direction));
- }
-
- private:
-
- double get_value (const double& r) const
- {
- // save a copy of this value for later
- if (scalar_r)
- *scalar_r = r;
-
- return r;
- }
-
- template <typename U>
- double get_value (const U& r) const
- {
- // U should be a matrix type
- COMPILE_TIME_ASSERT(is_matrix<U>::value);
-
- // save a copy of this value for later
- if (matrix_r)
- *matrix_r = r;
-
- return dot(r,direction);
- }
-
- const funct& f;
- const T& start;
- const T& direction;
- T* matrix_r;
- double* scalar_r;
- };
-
- template <typename funct, typename T>
- const line_search_funct<funct,T> make_line_search_function(const funct& f, const T& start, const T& direction)
- {
- COMPILE_TIME_ASSERT(is_matrix<T>::value);
- DLIB_ASSERT (
- is_col_vector(start) && is_col_vector(direction) && start.size() == direction.size(),
- "\tline_search_funct make_line_search_function(f,start,direction)"
- << "\n\tYou have to supply column vectors to this function"
- << "\n\tstart.nc(): " << start.nc()
- << "\n\tdirection.nc(): " << direction.nc()
- << "\n\tstart.nr(): " << start.nr()
- << "\n\tdirection.nr(): " << direction.nr()
- );
- return line_search_funct<funct,T>(f,start,direction);
- }
-
-// ----------------------------------------------------------------------------------------
-
- template <typename funct, typename T>
- const line_search_funct<funct,T> make_line_search_function(const funct& f, const T& start, const T& direction, double& f_out)
- {
- COMPILE_TIME_ASSERT(is_matrix<T>::value);
- DLIB_ASSERT (
- is_col_vector(start) && is_col_vector(direction) && start.size() == direction.size(),
- "\tline_search_funct make_line_search_function(f,start,direction)"
- << "\n\tYou have to supply column vectors to this function"
- << "\n\tstart.nc(): " << start.nc()
- << "\n\tdirection.nc(): " << direction.nc()
- << "\n\tstart.nr(): " << start.nr()
- << "\n\tdirection.nr(): " << direction.nr()
- );
- return line_search_funct<funct,T>(f,start,direction, f_out);
- }
-
-// ----------------------------------------------------------------------------------------
-
- template <typename funct, typename T>
- const line_search_funct<funct,T> make_line_search_function(const funct& f, const T& start, const T& direction, T& grad_out)
- {
- COMPILE_TIME_ASSERT(is_matrix<T>::value);
- DLIB_ASSERT (
- is_col_vector(start) && is_col_vector(direction) && start.size() == direction.size(),
- "\tline_search_funct make_line_search_function(f,start,direction)"
- << "\n\tYou have to supply column vectors to this function"
- << "\n\tstart.nc(): " << start.nc()
- << "\n\tdirection.nc(): " << direction.nc()
- << "\n\tstart.nr(): " << start.nr()
- << "\n\tdirection.nr(): " << direction.nr()
- );
- return line_search_funct<funct,T>(f,start,direction,grad_out);
- }
-
-// ----------------------------------------------------------------------------------------
-// ----------------------------------------------------------------------------------------
-// ----------------------------------------------------------------------------------------
-
- inline double poly_min_extrap (
- double f0,
- double d0,
- double f1,
- double d1,
- double limit = 1
- )
- {
- const double n = 3*(f1 - f0) - 2*d0 - d1;
- const double e = d0 + d1 - 2*(f1 - f0);
-
-
- // find the minimum of the derivative of the polynomial
-
- double temp = std::max(n*n - 3*e*d0,0.0);
-
- if (temp < 0)
- return 0.5;
-
- temp = std::sqrt(temp);
-
- if (std::abs(e) <= std::numeric_limits<double>::epsilon())
- return 0.5;
-
- // figure out the two possible min values
- double x1 = (temp - n)/(3*e);
- double x2 = -(temp + n)/(3*e);
-
- // compute the value of the interpolating polynomial at these two points
- double y1 = f0 + d0*x1 + n*x1*x1 + e*x1*x1*x1;
- double y2 = f0 + d0*x2 + n*x2*x2 + e*x2*x2*x2;
-
- // pick the best point
- double x;
- if (y1 < y2)
- x = x1;
- else
- x = x2;
-
- // now make sure the minimum is within the allowed range of [0,limit]
- return put_in_range(0,limit,x);
- }
-
-// ----------------------------------------------------------------------------------------
-
- inline double poly_min_extrap (
- double f0,
- double d0,
- double f1
- )
- {
- const double temp = 2*(f1 - f0 - d0);
- if (std::abs(temp) <= d0*std::numeric_limits<double>::epsilon())
- return 0.5;
-
- const double alpha = -d0/temp;
-
- // now make sure the minimum is within the allowed range of (0,1)
- return put_in_range(0,1,alpha);
- }
-
-// ----------------------------------------------------------------------------------------
-
- inline double poly_min_extrap (
- double f0,
- double d0,
- double x1,
- double f_x1,
- double x2,
- double f_x2
- )
- {
- DLIB_ASSERT(0 < x1 && x1 < x2,"Invalid inputs were given to this function");
- // The contents of this function follow the equations described on page 58 of the
- // book Numerical Optimization by Nocedal and Wright, second edition.
- matrix<double,2,2> m;
- matrix<double,2,1> v;
-
- const double aa2 = x2*x2;
- const double aa1 = x1*x1;
- m = aa2, -aa1,
- -aa2*x2, aa1*x1;
- v = f_x1 - f0 - d0*x1,
- f_x2 - f0 - d0*x2;
-
-
- double temp = aa2*aa1*(x1-x2);
-
- // just take a guess if this happens
- if (temp == 0 || std::fpclassify(temp) == FP_SUBNORMAL)
- {
- return x1/2.0;
- }
-
- matrix<double,2,1> temp2;
- temp2 = m*v/temp;
- const double a = temp2(0);
- const double b = temp2(1);
-
- temp = b*b - 3*a*d0;
- if (temp < 0 || a == 0)
- {
- // This is probably a line so just pick the lowest point
- if (f0 < f_x2)
- return 0;
- else
- return x2;
- }
- temp = (-b + std::sqrt(temp))/(3*a);
- return put_in_range(0, x2, temp);
- }
-
-// ----------------------------------------------------------------------------------------
-
- inline double lagrange_poly_min_extrap (
- double p1,
- double p2,
- double p3,
- double f1,
- double f2,
- double f3
- )
- {
- DLIB_ASSERT(p1 < p2 && p2 < p3 && f1 >= f2 && f2 <= f3,
- " p1: " << p1
- << " p2: " << p2
- << " p3: " << p3
- << " f1: " << f1
- << " f2: " << f2
- << " f3: " << f3);
-
- // This formula is out of the book Nonlinear Optimization by Andrzej Ruszczynski. See section 5.2.
- double temp1 = f1*(p3*p3 - p2*p2) + f2*(p1*p1 - p3*p3) + f3*(p2*p2 - p1*p1);
- double temp2 = 2*(f1*(p3 - p2) + f2*(p1 - p3) + f3*(p2 - p1) );
-
- if (temp2 == 0)
- {
- return p2;
- }
-
- const double result = temp1/temp2;
-
- // do a final sanity check to make sure the result is in the right range
- if (p1 <= result && result <= p3)
- {
- return result;
- }
- else
- {
- return std::min(std::max(p1,result),p3);
- }
- }
-
-// ----------------------------------------------------------------------------------------
-
- template <
- typename funct,
- typename funct_der
- >
- double line_search (
- const funct& f,
- const double f0,
- const funct_der& der,
- const double d0,
- double rho,
- double sigma,
- double min_f,
- unsigned long max_iter
- )
- {
- DLIB_ASSERT (
- 0 < rho && rho < sigma && sigma < 1 && max_iter > 0,
- "\tdouble line_search()"
- << "\n\tYou have given invalid arguments to this function"
- << "\n\t sigma: " << sigma
- << "\n\t rho: " << rho
- << "\n\t max_iter: " << max_iter
- );
-
- // The bracketing phase of this function is implemented according to block 2.6.2 from
- // the book Practical Methods of Optimization by R. Fletcher. The sectioning
- // phase is an implementation of 2.6.4 from the same book.
-
- // 1 <= tau1a < tau1b. Controls the alpha jump size during the bracketing phase of
- // the search.
- const double tau1a = 1.4;
- const double tau1b = 9;
-
- // it must be the case that 0 < tau2 < tau3 <= 1/2 for the algorithm to function
- // correctly but the specific values of tau2 and tau3 aren't super important.
- const double tau2 = 1.0/10.0;
- const double tau3 = 1.0/2.0;
-
-
- // Stop right away and return a step size of 0 if the gradient is 0 at the starting point
- if (std::abs(d0) <= std::abs(f0)*std::numeric_limits<double>::epsilon())
- return 0;
-
- // Stop right away if the current value is good enough according to min_f
- if (f0 <= min_f)
- return 0;
-
- // Figure out a reasonable upper bound on how large alpha can get.
- const double mu = (min_f-f0)/(rho*d0);
-
-
- double alpha = 1;
- if (mu < 0)
- alpha = -alpha;
- alpha = put_in_range(0, 0.65*mu, alpha);
-
-
- double last_alpha = 0;
- double last_val = f0;
- double last_val_der = d0;
-
- // The bracketing stage will find a range of points [a,b]
- // that contains a reasonable solution to the line search
- double a, b;
-
- // These variables will hold the values and derivatives of f(a) and f(b)
- double a_val, b_val, a_val_der, b_val_der;
-
- // This thresh value represents the Wolfe curvature condition
- const double thresh = std::abs(sigma*d0);
-
- unsigned long itr = 0;
- // do the bracketing stage to find the bracket range [a,b]
- while (true)
- {
- ++itr;
- const double val = f(alpha);
- const double val_der = der(alpha);
-
- // we are done with the line search since we found a value smaller
- // than the minimum f value
- if (val <= min_f)
- return alpha;
-
- if (val > f0 + rho*alpha*d0 || val >= last_val)
- {
- a_val = last_val;
- a_val_der = last_val_der;
- b_val = val;
- b_val_der = val_der;
-
- a = last_alpha;
- b = alpha;
- break;
- }
-
- if (std::abs(val_der) <= thresh)
- return alpha;
-
- // if we are stuck not making progress then quit with the current alpha
- if (last_alpha == alpha || itr >= max_iter)
- return alpha;
-
- if (val_der >= 0)
- {
- a_val = val;
- a_val_der = val_der;
- b_val = last_val;
- b_val_der = last_val_der;
-
- a = alpha;
- b = last_alpha;
- break;
- }
-
-
-
- const double temp = alpha;
- // Pick a larger range [first, last]. We will pick the next alpha in that
- // range.
- double first, last;
- if (mu > 0)
- {
- first = std::min(mu, alpha + tau1a*(alpha - last_alpha));
- last = std::min(mu, alpha + tau1b*(alpha - last_alpha));
- }
- else
- {
- first = std::max(mu, alpha + tau1a*(alpha - last_alpha));
- last = std::max(mu, alpha + tau1b*(alpha - last_alpha));
- }
-
-
-
- // pick a point between first and last by doing some kind of interpolation
- if (last_alpha < alpha)
- alpha = last_alpha + (alpha-last_alpha)*poly_min_extrap(last_val, last_val_der, val, val_der, 1e10);
- else
- alpha = alpha + (last_alpha-alpha)*poly_min_extrap(val, val_der, last_val, last_val_der, 1e10);
-
- alpha = put_in_range(first,last,alpha);
-
- last_alpha = temp;
-
- last_val = val;
- last_val_der = val_der;
-
- }
-
-
- // Now do the sectioning phase from 2.6.4
- while (true)
- {
- ++itr;
- double first = a + tau2*(b-a);
- double last = b - tau3*(b-a);
-
- // use interpolation to pick alpha between first and last
- alpha = a + (b-a)*poly_min_extrap(a_val, a_val_der, b_val, b_val_der);
- alpha = put_in_range(first,last,alpha);
-
- const double val = f(alpha);
- const double val_der = der(alpha);
-
- // we are done with the line search since we found a value smaller
- // than the minimum f value or we ran out of iterations.
- if (val <= min_f || itr >= max_iter)
- return alpha;
-
- // stop if the interval gets so small that it isn't shrinking any more due to rounding error
- if (a == first || b == last)
- {
- return b;
- }
-
- // If alpha has basically become zero then just stop. Think of it like this,
- // if we take the largest possible alpha step will the objective function
- // change at all? If not then there isn't any point looking for a better
- // alpha.
- const double max_possible_alpha = std::max(std::abs(a),std::abs(b));
- if (std::abs(max_possible_alpha*d0) <= std::abs(f0)*std::numeric_limits<double>::epsilon())
- return alpha;
-
-
- if (val > f0 + rho*alpha*d0 || val >= a_val)
- {
- b = alpha;
- b_val = val;
- b_val_der = val_der;
- }
- else
- {
- if (std::abs(val_der) <= thresh)
- return alpha;
-
- if ( (b-a)*val_der >= 0)
- {
- b = a;
- b_val = a_val;
- b_val_der = a_val_der;
- }
-
- a = alpha;
- a_val = val;
- a_val_der = val_der;
- }
- }
- }
-
-// ----------------------------------------------------------------------------------------
-
- template <
- typename funct
- >
- double backtracking_line_search (
- const funct& f,
- double f0,
- double d0,
- double alpha,
- double rho,
- unsigned long max_iter
- )
- {
- DLIB_ASSERT (
- 0 < rho && rho < 1 && max_iter > 0,
- "\tdouble backtracking_line_search()"
- << "\n\tYou have given invalid arguments to this function"
- << "\n\t rho: " << rho
- << "\n\t max_iter: " << max_iter
- );
-
- // make sure alpha is going in the right direction. That is, it should be opposite
- // the direction of the gradient.
- if ((d0 > 0 && alpha > 0) ||
- (d0 < 0 && alpha < 0))
- {
- alpha *= -1;
- }
-
- bool have_prev_alpha = false;
- double prev_alpha = 0;
- double prev_val = 0;
- unsigned long iter = 0;
- while (true)
- {
- ++iter;
- const double val = f(alpha);
- if (val <= f0 + alpha*rho*d0 || iter >= max_iter)
- {
- return alpha;
- }
- else
- {
- // Interpolate a new alpha. We also make sure the step by which we
- // reduce alpha is not super small.
- double step;
- if (!have_prev_alpha)
- {
- if (d0 < 0)
- step = alpha*put_in_range(0.1,0.9, poly_min_extrap(f0, d0, val));
- else
- step = alpha*put_in_range(0.1,0.9, poly_min_extrap(f0, -d0, val));
- have_prev_alpha = true;
- }
- else
- {
- if (d0 < 0)
- step = put_in_range(0.1*alpha,0.9*alpha, poly_min_extrap(f0, d0, alpha, val, prev_alpha, prev_val));
- else
- step = put_in_range(0.1*alpha,0.9*alpha, -poly_min_extrap(f0, -d0, -alpha, val, -prev_alpha, prev_val));
- }
-
- prev_alpha = alpha;
- prev_val = val;
-
- alpha = step;
- }
- }
- }
-
-// ----------------------------------------------------------------------------------------
-
- class optimize_single_variable_failure : public error {
- public: optimize_single_variable_failure(const std::string& s):error(s){}
- };
-
-// ----------------------------------------------------------------------------------------
-
- template <typename funct>
- double find_min_single_variable (
- const funct& f,
- double& starting_point,
- const double begin = -1e200,
- const double end = 1e200,
- const double eps = 1e-3,
- const long max_iter = 100,
- const double initial_search_radius = 1
- )
- {
- DLIB_CASSERT( eps > 0 &&
- max_iter > 1 &&
- begin <= starting_point && starting_point <= end &&
- initial_search_radius > 0,
- "eps: " << eps
- << "\n max_iter: "<< max_iter
- << "\n begin: "<< begin
- << "\n end: "<< end
- << "\n starting_point: "<< starting_point
- << "\n initial_search_radius: "<< initial_search_radius
- );
-
- double search_radius = initial_search_radius;
-
- double p1=0, p2=0, p3=0, f1=0, f2=0, f3=0;
- long f_evals = 1;
-
- if (begin == end)
- {
- return f(starting_point);
- }
-
- using std::abs;
- using std::min;
- using std::max;
-
- // find three bracketing points such that f1 > f2 < f3. Do this by generating a sequence
- // of points expanding away from 0. Also note that, in the following code, it is always the
- // case that p1 < p2 < p3.
-
-
-
- // The first thing we do is get a starting set of 3 points that are inside the [begin,end] bounds
- p1 = max(starting_point-search_radius, begin);
- p3 = min(starting_point+search_radius, end);
- f1 = f(p1);
- f3 = f(p3);
-
- if (starting_point == p1 || starting_point == p3)
- {
- p2 = (p1+p3)/2;
- f2 = f(p2);
- }
- else
- {
- p2 = starting_point;
- f2 = f(starting_point);
- }
-
- f_evals += 2;
-
- // Now we have 3 points on the function. Start looking for a bracketing set such that
- // f1 > f2 < f3 is the case.
- while ( !(f1 > f2 && f2 < f3))
- {
- // check for hitting max_iter or if the interval is now too small
- if (f_evals >= max_iter)
- {
- throw optimize_single_variable_failure(
- "The max number of iterations of single variable optimization have been reached\n"
- "without converging.");
- }
- if (p3-p1 < eps)
- {
- if (f1 < min(f2,f3))
- {
- starting_point = p1;
- return f1;
- }
-
- if (f2 < min(f1,f3))
- {
- starting_point = p2;
- return f2;
- }
-
- starting_point = p3;
- return f3;
- }
-
- // If the left most points are identical in function value then expand out the
- // left a bit, unless it's already at bound or we would drop that left most
- // point anyway because it's bad.
- if (f1==f2 && f1<f3 && p1!=begin)
- {
- p1 = max(p1 - search_radius, begin);
- f1 = f(p1);
- ++f_evals;
- search_radius *= 2;
- continue;
- }
- if (f2==f3 && f3<f1 && p3!=end)
- {
- p3 = min(p3 + search_radius, end);
- f3 = f(p3);
- ++f_evals;
- search_radius *= 2;
- continue;
- }
-
-
- // if f1 is small then take a step to the left
- if (f1 <= f3)
- {
- // check if the minimum is butting up against the bounds and if so then pick
- // a point between p1 and p2 in the hopes that shrinking the interval will
- // be a good thing to do. Or if p1 and p2 aren't differentiated then try and
- // get them to obtain different values.
- if (p1 == begin || (f1 == f2 && (end-begin) < search_radius ))
- {
- p3 = p2;
- f3 = f2;
-
- p2 = (p1+p2)/2.0;
- f2 = f(p2);
- }
- else
- {
- // pick a new point to the left of our current bracket
- p3 = p2;
- f3 = f2;
-
- p2 = p1;
- f2 = f1;
-
- p1 = max(p1 - search_radius, begin);
- f1 = f(p1);
-
- search_radius *= 2;
- }
-
- }
- // otherwise f3 is small and we should take a step to the right
- else
- {
- // check if the minimum is butting up against the bounds and if so then pick
- // a point between p2 and p3 in the hopes that shrinking the interval will
- // be a good thing to do. Or if p2 and p3 aren't differentiated then try and
- // get them to obtain different values.
- if (p3 == end || (f2 == f3 && (end-begin) < search_radius))
- {
- p1 = p2;
- f1 = f2;
-
- p2 = (p3+p2)/2.0;
- f2 = f(p2);
- }
- else
- {
- // pick a new point to the right of our current bracket
- p1 = p2;
- f1 = f2;
-
- p2 = p3;
- f2 = f3;
-
- p3 = min(p3 + search_radius, end);
- f3 = f(p3);
-
- search_radius *= 2;
- }
- }
-
- ++f_evals;
- }
-
-
- // Loop until we have done the max allowable number of iterations or
- // the bracketing window is smaller than eps.
- // Within this loop we maintain the invariant that: f1 > f2 < f3 and p1 < p2 < p3
- const double tau = 0.1;
- while( f_evals < max_iter && p3-p1 > eps)
- {
- double p_min = lagrange_poly_min_extrap(p1,p2,p3, f1,f2,f3);
-
-
- // make sure p_min isn't too close to the three points we already have
- if (p_min < p2)
- {
- const double min_dist = (p2-p1)*tau;
- if (abs(p1-p_min) < min_dist)
- {
- p_min = p1 + min_dist;
- }
- else if (abs(p2-p_min) < min_dist)
- {
- p_min = p2 - min_dist;
- }
- }
- else
- {
- const double min_dist = (p3-p2)*tau;
- if (abs(p2-p_min) < min_dist)
- {
- p_min = p2 + min_dist;
- }
- else if (abs(p3-p_min) < min_dist)
- {
- p_min = p3 - min_dist;
- }
- }
-
- // make sure one side of the bracket isn't super huge compared to the other
- // side. If it is then contract it.
- const double bracket_ratio = abs(p1-p2)/abs(p2-p3);
- if ( !( bracket_ratio < 10 && bracket_ratio > 0.1) )
- {
- // Force p_min to be on a reasonable side. But only if lagrange_poly_min_extrap()
- // didn't put it on a good side already.
- if (bracket_ratio > 1 && p_min > p2)
- p_min = (p1+p2)/2;
- else if (p_min < p2)
- p_min = (p2+p3)/2;
- }
-
-
- const double f_min = f(p_min);
-
-
- // Remove one of the endpoints of our bracket depending on where the new point falls.
- if (p_min < p2)
- {
- if (f1 > f_min && f_min < f2)
- {
- p3 = p2;
- f3 = f2;
- p2 = p_min;
- f2 = f_min;
- }
- else
- {
- p1 = p_min;
- f1 = f_min;
- }
- }
- else
- {
- if (f2 > f_min && f_min < f3)
- {
- p1 = p2;
- f1 = f2;
- p2 = p_min;
- f2 = f_min;
- }
- else
- {
- p3 = p_min;
- f3 = f_min;
- }
- }
-
-
- ++f_evals;
- }
-
- if (f_evals >= max_iter)
- {
- throw optimize_single_variable_failure(
- "The max number of iterations of single variable optimization have been reached\n"
- "without converging.");
- }
-
- starting_point = p2;
- return f2;
- }
-
-// ----------------------------------------------------------------------------------------
-
- template <typename funct>
- class negate_function_object
- {
- public:
- negate_function_object(const funct& f_) : f(f_){}
-
- template <typename T>
- double operator()(const T& x) const
- {
- return -f(x);
- }
-
- private:
- const funct& f;
- };
-
- template <typename funct>
- const negate_function_object<funct> negate_function(const funct& f) { return negate_function_object<funct>(f); }
-
-// ----------------------------------------------------------------------------------------
-
- template <typename funct>
- double find_max_single_variable (
- const funct& f,
- double& starting_point,
- const double begin = -1e200,
- const double end = 1e200,
- const double eps = 1e-3,
- const long max_iter = 100,
- const double initial_search_radius = 1
- )
- {
- return -find_min_single_variable(negate_function(f), starting_point, begin, end, eps, max_iter, initial_search_radius);
- }
-
-// ----------------------------------------------------------------------------------------
-
-}
-
-#endif // DLIB_OPTIMIZATIOn_LINE_SEARCH_H_
-
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