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diff --git a/src/ml/dlib/examples/optimization_ex.cpp b/src/ml/dlib/examples/optimization_ex.cpp new file mode 100644 index 000000000..2d35fa814 --- /dev/null +++ b/src/ml/dlib/examples/optimization_ex.cpp @@ -0,0 +1,319 @@ +// The contents of this file are in the public domain. See LICENSE_FOR_EXAMPLE_PROGRAMS.txt +/* + + This is an example illustrating the use the general purpose non-linear + optimization routines from the dlib C++ Library. + + The library provides implementations of many popular algorithms such as L-BFGS + and BOBYQA. These algorithms allow you to find the minimum or maximum of a + function of many input variables. This example walks though a few of the ways + you might put these routines to use. + +*/ + + +#include <dlib/optimization.h> +#include <dlib/global_optimization.h> +#include <iostream> + + +using namespace std; +using namespace dlib; + +// ---------------------------------------------------------------------------------------- + +// In dlib, most of the general purpose solvers optimize functions that take a +// column vector as input and return a double. So here we make a typedef for a +// variable length column vector of doubles. This is the type we will use to +// represent the input to our objective functions which we will be minimizing. +typedef matrix<double,0,1> column_vector; + +// ---------------------------------------------------------------------------------------- +// Below we create a few functions. When you get down into main() you will see that +// we can use the optimization algorithms to find the minimums of these functions. +// ---------------------------------------------------------------------------------------- + +double rosen (const column_vector& m) +/* + This function computes what is known as Rosenbrock's function. It is + a function of two input variables and has a global minimum at (1,1). + So when we use this function to test out the optimization algorithms + we will see that the minimum found is indeed at the point (1,1). +*/ +{ + const double x = m(0); + const double y = m(1); + + // compute Rosenbrock's function and return the result + return 100.0*pow(y - x*x,2) + pow(1 - x,2); +} + +// This is a helper function used while optimizing the rosen() function. +const column_vector rosen_derivative (const column_vector& m) +/*! + ensures + - returns the gradient vector for the rosen function +!*/ +{ + const double x = m(0); + const double y = m(1); + + // make us a column vector of length 2 + column_vector res(2); + + // now compute the gradient vector + res(0) = -400*x*(y-x*x) - 2*(1-x); // derivative of rosen() with respect to x + res(1) = 200*(y-x*x); // derivative of rosen() with respect to y + return res; +} + +// This function computes the Hessian matrix for the rosen() fuction. This is +// the matrix of second derivatives. +matrix<double> rosen_hessian (const column_vector& m) +{ + const double x = m(0); + const double y = m(1); + + matrix<double> res(2,2); + + // now compute the second derivatives + res(0,0) = 1200*x*x - 400*y + 2; // second derivative with respect to x + res(1,0) = res(0,1) = -400*x; // derivative with respect to x and y + res(1,1) = 200; // second derivative with respect to y + return res; +} + +// ---------------------------------------------------------------------------------------- + +class rosen_model +{ + /*! + This object is a "function model" which can be used with the + find_min_trust_region() routine. + !*/ + +public: + typedef ::column_vector column_vector; + typedef matrix<double> general_matrix; + + double operator() ( + const column_vector& x + ) const { return rosen(x); } + + void get_derivative_and_hessian ( + const column_vector& x, + column_vector& der, + general_matrix& hess + ) const + { + der = rosen_derivative(x); + hess = rosen_hessian(x); + } +}; + +// ---------------------------------------------------------------------------------------- + +int main() try +{ + // Set the starting point to (4,8). This is the point the optimization algorithm + // will start out from and it will move it closer and closer to the function's + // minimum point. So generally you want to try and compute a good guess that is + // somewhat near the actual optimum value. + column_vector starting_point = {4, 8}; + + // The first example below finds the minimum of the rosen() function and uses the + // analytical derivative computed by rosen_derivative(). Since it is very easy to + // make a mistake while coding a function like rosen_derivative() it is a good idea + // to compare your derivative function against a numerical approximation and see if + // the results are similar. If they are very different then you probably made a + // mistake. So the first thing we do is compare the results at a test point: + cout << "Difference between analytic derivative and numerical approximation of derivative: " + << length(derivative(rosen)(starting_point) - rosen_derivative(starting_point)) << endl; + + + cout << "Find the minimum of the rosen function()" << endl; + // Now we use the find_min() function to find the minimum point. The first argument + // to this routine is the search strategy we want to use. The second argument is the + // stopping strategy. Below I'm using the objective_delta_stop_strategy which just + // says that the search should stop when the change in the function being optimized + // is small enough. + + // The other arguments to find_min() are the function to be minimized, its derivative, + // then the starting point, and the last is an acceptable minimum value of the rosen() + // function. That is, if the algorithm finds any inputs to rosen() that gives an output + // value <= -1 then it will stop immediately. Usually you supply a number smaller than + // the actual global minimum. So since the smallest output of the rosen function is 0 + // we just put -1 here which effectively causes this last argument to be disregarded. + + find_min(bfgs_search_strategy(), // Use BFGS search algorithm + objective_delta_stop_strategy(1e-7), // Stop when the change in rosen() is less than 1e-7 + rosen, rosen_derivative, starting_point, -1); + // Once the function ends the starting_point vector will contain the optimum point + // of (1,1). + cout << "rosen solution:\n" << starting_point << endl; + + + // Now let's try doing it again with a different starting point and the version + // of find_min() that doesn't require you to supply a derivative function. + // This version will compute a numerical approximation of the derivative since + // we didn't supply one to it. + starting_point = {-94, 5.2}; + find_min_using_approximate_derivatives(bfgs_search_strategy(), + objective_delta_stop_strategy(1e-7), + rosen, starting_point, -1); + // Again the correct minimum point is found and stored in starting_point + cout << "rosen solution:\n" << starting_point << endl; + + + // Here we repeat the same thing as above but this time using the L-BFGS + // algorithm. L-BFGS is very similar to the BFGS algorithm, however, BFGS + // uses O(N^2) memory where N is the size of the starting_point vector. + // The L-BFGS algorithm however uses only O(N) memory. So if you have a + // function of a huge number of variables the L-BFGS algorithm is probably + // a better choice. + starting_point = {0.8, 1.3}; + find_min(lbfgs_search_strategy(10), // The 10 here is basically a measure of how much memory L-BFGS will use. + objective_delta_stop_strategy(1e-7).be_verbose(), // Adding be_verbose() causes a message to be + // printed for each iteration of optimization. + rosen, rosen_derivative, starting_point, -1); + + cout << endl << "rosen solution: \n" << starting_point << endl; + + starting_point = {-94, 5.2}; + find_min_using_approximate_derivatives(lbfgs_search_strategy(10), + objective_delta_stop_strategy(1e-7), + rosen, starting_point, -1); + cout << "rosen solution: \n"<< starting_point << endl; + + + + + // dlib also supports solving functions subject to bounds constraints on + // the variables. So for example, if you wanted to find the minimizer + // of the rosen function where both input variables were in the range + // 0.1 to 0.8 you would do it like this: + starting_point = {0.1, 0.1}; // Start with a valid point inside the constraint box. + find_min_box_constrained(lbfgs_search_strategy(10), + objective_delta_stop_strategy(1e-9), + rosen, rosen_derivative, starting_point, 0.1, 0.8); + // Here we put the same [0.1 0.8] range constraint on each variable, however, you + // can put different bounds on each variable by passing in column vectors of + // constraints for the last two arguments rather than scalars. + + cout << endl << "constrained rosen solution: \n" << starting_point << endl; + + // You can also use an approximate derivative like so: + starting_point = {0.1, 0.1}; + find_min_box_constrained(bfgs_search_strategy(), + objective_delta_stop_strategy(1e-9), + rosen, derivative(rosen), starting_point, 0.1, 0.8); + cout << endl << "constrained rosen solution: \n" << starting_point << endl; + + + + + // In many cases, it is useful if we also provide second derivative information + // to the optimizers. Two examples of how we can do that are shown below. + starting_point = {0.8, 1.3}; + find_min(newton_search_strategy(rosen_hessian), + objective_delta_stop_strategy(1e-7), + rosen, + rosen_derivative, + starting_point, + -1); + cout << "rosen solution: \n"<< starting_point << endl; + + // We can also use find_min_trust_region(), which is also a method which uses + // second derivatives. For some kinds of non-convex function it may be more + // reliable than using a newton_search_strategy with find_min(). + starting_point = {0.8, 1.3}; + find_min_trust_region(objective_delta_stop_strategy(1e-7), + rosen_model(), + starting_point, + 10 // initial trust region radius + ); + cout << "rosen solution: \n"<< starting_point << endl; + + + + + + // Next, let's try the BOBYQA algorithm. This is a technique specially + // designed to minimize a function in the absence of derivative information. + // Generally speaking, it is the method of choice if derivatives are not available + // and the function you are optimizing is smooth and has only one local optima. As + // an example, consider the be_like_target function defined below: + column_vector target = {3, 5, 1, 7}; + auto be_like_target = [&](const column_vector& x) { + return mean(squared(x-target)); + }; + starting_point = {-4,5,99,3}; + find_min_bobyqa(be_like_target, + starting_point, + 9, // number of interpolation points + uniform_matrix<double>(4,1, -1e100), // lower bound constraint + uniform_matrix<double>(4,1, 1e100), // upper bound constraint + 10, // initial trust region radius + 1e-6, // stopping trust region radius + 100 // max number of objective function evaluations + ); + cout << "be_like_target solution:\n" << starting_point << endl; + + + + + + // Finally, let's try the find_min_global() routine. Like find_min_bobyqa(), + // this technique is specially designed to minimize a function in the absence + // of derivative information. However, it is also designed to handle + // functions with many local optima. Where BOBYQA would get stuck at the + // nearest local optima, find_min_global() won't. find_min_global() uses a + // global optimization method based on a combination of non-parametric global + // function modeling and BOBYQA style quadratic trust region modeling to + // efficiently find a global minimizer. It usually does a good job with a + // relatively small number of calls to the function being optimized. + // + // You also don't have to give it a starting point or set any parameters, + // other than defining bounds constraints. This makes it the method of + // choice for derivative free optimization in the presence of multiple local + // optima. Its API also allows you to define functions that take a + // column_vector as shown above or to explicitly use named doubles as + // arguments, which we do here. + auto complex_holder_table = [](double x0, double x1) + { + // This function is a version of the well known Holder table test + // function, which is a function containing a bunch of local optima. + // Here we make it even more difficult by adding more local optima + // and also a bunch of discontinuities. + + // add discontinuities + double sign = 1; + for (double j = -4; j < 9; j += 0.5) + { + if (j < x0 && x0 < j+0.5) + x0 += sign*0.25; + sign *= -1; + } + // Holder table function tilted towards 10,10 and with additional + // high frequency terms to add more local optima. + return -( std::abs(sin(x0)*cos(x1)*exp(std::abs(1-std::sqrt(x0*x0+x1*x1)/pi))) -(x0+x1)/10 - sin(x0*10)*cos(x1*10)); + }; + + // To optimize this difficult function all we need to do is call + // find_min_global() + auto result = find_min_global(complex_holder_table, + {-10,-10}, // lower bounds + {10,10}, // upper bounds + max_function_calls(300)); + + cout.precision(9); + // These cout statements will show that find_min_global() found the + // globally optimal solution to 9 digits of precision: + cout << "complex holder table function solution y (should be -21.9210397): " << result.y << endl; + cout << "complex holder table function solution x:\n" << result.x << endl; +} +catch (std::exception& e) +{ + cout << e.what() << endl; +} + |