From 476a0f05d18b3bbefdaf371a9160216fb38f5574 Mon Sep 17 00:00:00 2001 From: Amit Rotem Date: Thu, 10 Jan 2019 11:19:51 -0700 Subject: [PATCH] updates to roots finding and optimization --- ODEs/ODE.hpp | 18 ++++- ODEs/nonlin_bvp.cpp | 8 +- bfgs.cpp | 18 ----- broyd.cpp | 3 + cyc_queue.cpp | 21 ++++- examples/bvp_ex.cpp | 13 ++- examples/make_bvp_ex | 26 ++++++ examples/make_minimize_ex | 4 +- examples/minimize_ex.cpp | 40 ++++++--- examples/newton_ex.cpp | 10 ++- examples/roots_ex.cpp | 19 +++-- finite_dif.cpp | 7 ++ fzero.cpp | 165 ++++++++++++++++++++++++++++++++++++++ line_min.cpp | 66 +++++++++++++++ minimize_unc.cpp | 16 ++-- momentum.cpp | 18 +++-- nlcgd.cpp | 80 ++++++------------ numerics.hpp | 63 +++++++++++---- pois_ex | Bin 2328224 -> 0 bytes roots.cpp | 92 --------------------- sgd.cpp | 36 +++++---- statistics/statistics.hpp | 14 ++++ wolfe_step.cpp | 18 +++++ 23 files changed, 497 insertions(+), 258 deletions(-) create mode 100644 examples/make_bvp_ex create mode 100755 fzero.cpp create mode 100644 line_min.cpp delete mode 100755 pois_ex delete mode 100755 roots.cpp create mode 100644 wolfe_step.cpp diff --git a/ODEs/ODE.hpp b/ODEs/ODE.hpp index d9c79ce..68e88fd 100644 --- a/ODEs/ODE.hpp +++ b/ODEs/ODE.hpp @@ -2,6 +2,20 @@ #include "../numerics.hpp" +/* Copyright 2019 Amit Rotem + +Licensed under the Apache License, Version 2.0 (the "License"); +you may not use this file except in compliance with the License. +You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + +Unless required by applicable law or agreed to in writing, software +distributed under the License is distributed on an "AS IS" BASIS, +WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +See the License for the specific language governing permissions and +limitations under the License. */ + namespace ODE { // --- ode solver constants --- // const double rk45_kmin = 1.0e-4; @@ -194,8 +208,8 @@ namespace ODE { arma::vec am2(std::function, arma::vec&, double, ivp_options&); arma::vec am2(std::function, arma::vec&, double); - numerics::CubicInterp IVP_solve(const odefun&, arma::vec&, arma::mat&, ivp_options&, ode_solver solver = RK45); - numerics::CubicInterp IVP_solve(const odefun&, arma::vec&, arma::mat&, ode_solver solver = RK45); + dsolnc IVP_solve(const odefun&, arma::vec&, arma::mat&, ivp_options&, ode_solver solver = RK45); + dsolnc IVP_solve(const odefun&, arma::vec&, arma::mat&, ode_solver solver = RK45); // --- BVPs ------------------- // class linear_BVP { //--- u''(x) = a(x) + b(x)*u(x) + c(x)*u'(x) ---// diff --git a/ODEs/nonlin_bvp.cpp b/ODEs/nonlin_bvp.cpp index c81ed34..04b774f 100644 --- a/ODEs/nonlin_bvp.cpp +++ b/ODEs/nonlin_bvp.cpp @@ -15,7 +15,7 @@ ODE::dsolnp ODE::bvp(const odefun& f, const bcfun& bc, const soln_init& guess, b arma::vec x; cheb(D, x, bc.xL, bc.xR, m-1); - auto ff = [&](const arma::vec& u) -> arma::vec { + numerics::vector_func ff = [&](const arma::vec& u) -> arma::vec { arma::mat U = arma::reshape(u,m,n); arma::rowvec BC = bc.func(u.row(0), u.row(m-1)); arma::mat A = D; @@ -30,7 +30,7 @@ ODE::dsolnp ODE::bvp(const odefun& f, const bcfun& bc, const soln_init& guess, b return z; }; // wrapper for root finding function - std::function J = [&](const arma::vec& u) -> arma::mat { + numerics::vec_mat_func J = [&](const arma::vec& u) -> arma::mat { arma::mat U = arma::reshape(u,m,n); arma::mat DF = arma::zeros(m*n,m*n); if (opts.jacobian_func != nullptr) { // Jacobian provided @@ -82,8 +82,8 @@ ODE::dsolnp ODE::bvp(const odefun& f, const bcfun& bc, const soln_init& guess, b opts.lsqropts.jacobian_func = &J; numerics::lmlsqr(ff, U0, opts.lsqropts); } else { // use Broyden solver - arma::mat J0 = J(U0); - opts.nlnopts.init_jacobian = &J0; + arma::mat JJ = J(U0); + opts.nlnopts.init_jacobian = &JJ; numerics::broyd(ff, U0, opts.nlnopts); } diff --git a/bfgs.cpp b/bfgs.cpp index 65122fb..20fbc4e 100644 --- a/bfgs.cpp +++ b/bfgs.cpp @@ -1,23 +1,5 @@ #include "numerics.hpp" -//--- rough approximator step size for quasi-newton methods based on strong wolfe conditions ---// -double numerics::wolfe_step(const vec_dfunc& obj_func, const vector_func& f, const arma::vec& x, const arma::vec& p, double c1, double c2, double b) { - double alpha = 1; - int k = 0; - while (true) { - if (k > 100) break; - double pfa = arma::dot(p, f(x + alpha*x)); - bool cond1 = obj_func(x + alpha*p) <= obj_func(x) + c1*alpha*pfa; - bool cond2 = std::abs(arma::dot(p, f(x + alpha*p))) <= c2*std::abs(arma::dot(p, f(x))); - if (cond1 && cond2) break; - else if (obj_func(x + b*alpha*p) < obj_func(x + alpha*p)) alpha *= b; - else alpha /= b; - k++; - } - return alpha; -} - - //--- Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm for local optimization ---// //----- f : vec = f(x) system of equations --------------------------------------// //----- x : initial guess close to a local minimum, root will be stored here -----// diff --git a/broyd.cpp b/broyd.cpp index bb937b3..6f67055 100644 --- a/broyd.cpp +++ b/broyd.cpp @@ -13,6 +13,9 @@ void numerics::broyd(const vector_func& f, arma::vec& x, nonlin_opts& opts) { else if (opts.init_jacobian != nullptr) { Jinv = *opts.init_jacobian; Jinv = arma::pinv(Jinv); + } else if (opts.jacobian_func != nullptr) { + Jinv = opts.jacobian_func->operator()(x); + Jinv = arma::pinv(Jinv); } else { // FD approximation always used to initialize. approx_jacobian(f,Jinv,x,1e-4); Jinv = arma::pinv(Jinv); diff --git a/cyc_queue.cpp b/cyc_queue.cpp index 975d0cb..0b813fa 100644 --- a/cyc_queue.cpp +++ b/cyc_queue.cpp @@ -11,7 +11,7 @@ numerics::cyc_queue::cyc_queue(size_t num_rows, size_t max_size) { } //--- add an element to the queue, removing first in element if the queue is full ---// -void numerics::cyc_queue::push(arma::vec& x) { +void numerics::cyc_queue::push(const arma::vec& x) { if (size < max_elem) { // queue not full A.col(size) = x; size++; @@ -26,13 +26,16 @@ arma::vec numerics::cyc_queue::operator()(size_t i) { if (i >= size) { std::cerr << "cyc_queue::element access out of bounds." << std::endl; return {0}; - } - else { + } else { int ind = mod(i + head, size); return A.col(ind); } } +arma::vec numerics::cyc_queue::end() { + return A.col(size-1); +} + //--- return current length of cyclic queue (relevent especially when not full) ---// int numerics::cyc_queue::length() { return size; @@ -41,4 +44,16 @@ int numerics::cyc_queue::length() { //--- return num_rows of the elements (each a vector) of the queue ---// int numerics::cyc_queue::col_size() { return A.n_rows; +} + +void numerics::cyc_queue::clear() { + A.fill(0); +} + +arma::mat numerics::cyc_queue::data() { + arma::mat D = arma::zeros(A.n_rows, size); + for (unsigned int i(0); i < size; ++i) { + D.col(i) = operator()(i); + } + return D; } \ No newline at end of file diff --git a/examples/bvp_ex.cpp b/examples/bvp_ex.cpp index 97b22eb..06c4eab 100644 --- a/examples/bvp_ex.cpp +++ b/examples/bvp_ex.cpp @@ -1,8 +1,6 @@ #include "../ODEs/ODE.hpp" #include "gnuplot_i.hpp" -// g++ -g -o bvp_ex polyInterp.cpp newton.cpp broyd.cpp lmlsqr.cpp finite_dif.cpp ODEs/cheb.cpp ODEs/linear_bvp.cpp ODEs/nonlin_bvp.cpp examples/bvp_ex.cpp examples/wait.cpp -larmadillo -lsuperlu - double afunc(double x) { return std::sin(x); } @@ -28,9 +26,9 @@ int main() { arma::vec x; arma::mat U; - // problem.solve(x,U,40); // 4th order FD approximation + problem.solve(x,U,40); // 4th order FD approximation // problem.solve(x,U,40, SECOND_ORDER); // 2nd order FD approximation - problem.solve(x,U,20, CHEBYSHEV); // spectral order approximation + // problem.solve(x,U,20, CHEBYSHEV); // spectral order approximation // dsolnp y = problem.solve(20); x = arma::linspace(0,2*M_PI); U = y.soln(x); // spectral order approximation outputing cheb polynomial arma::mat u = 0.25 * (arma::exp(-2*M_PI - x) % (-1 + 4*std::exp(2*M_PI)+arma::exp(2*x)) - 2*arma::sin(x)); @@ -73,7 +71,7 @@ int main() { bc.xR = 2*M_PI; bc.func = [](const arma::rowvec& uL, const arma::rowvec& uR) -> arma::rowvec { arma::rowvec v(2,arma::fill::zeros); - v(0) = uL(0) - 1; + v(0) = uL(0) - 1; // solution fixed as 1 at end points v(1) = uR(0) - 1; return v; }; @@ -87,8 +85,7 @@ int main() { bvp_opts opts; opts.num_points = 100; - opts.jacobian_func = &J; // providing a jacobian function improves runtime significantly - // opts.solver = numerics::LMLSQR; + // opts.jacobian_func = &J; // providing a jacobian function improves runtime significantly dsolnp soln = bvp(f, bc, guess, opts); x1 = arma::conv_to::from(soln.independent_var_values); @@ -96,7 +93,7 @@ int main() { stdv U2 = arma::conv_to::from(soln.solution_values.col(1)); graph.reset_plot(); - graph.set_style("points"); + graph.set_style("lines"); graph.plot_xy(x1,U1,"u(x) -- guess of cos(x)"); graph.plot_xy(x1,U2,"v(x) -- guess of sin(x)"); diff --git a/examples/make_bvp_ex b/examples/make_bvp_ex new file mode 100644 index 0000000..8a959fd --- /dev/null +++ b/examples/make_bvp_ex @@ -0,0 +1,26 @@ +CXX = g++ +CXXFLAGS = -g -Wall +LIBS = -larmadillo -lsuperlu +DEPS = ODEs/ODE.hpp +TARGET = bvp_ex + +all: bvp_ex + +bvp_ex: bvp_ex.o polyInterp.o wolfe_step.o newton.o broyd.o lmlsqr.o finite_dif.o cheb.o linear_bvp.o nonlin_bvp.o wait.o + $(CXX) $(CXXFLAGS) -o $(TARGET) *.o $(LIBS) + +%.o: %.cpp $(DEPS) +cheb.o: ODEs/cheb.cpp $(DEPS) + $(CXX) -g -Wall -c ODEs/cheb.cpp +linear_bvp.o: ODEs/linear_bvp.cpp $(DEPS) + $(CXX) -g -Wall -c ODEs/linear_bvp.cpp +nonlin_bvp.o: ODEs/nonlin_bvp.cpp $(DEPS) + $(CXX) -g -Wall -c ODEs/nonlin_bvp.cpp +bvp_ex.o: examples/bvp_ex.cpp $(DEPS) + $(CXX) -g -Wall -c examples/bvp_ex.cpp +wait.o: examples/wait.cpp + $(CXX) -g -Wall -c examples/wait.cpp + +clean: + rm *.o + rm bvp_ex \ No newline at end of file diff --git a/examples/make_minimize_ex b/examples/make_minimize_ex index 18561b7..31087ce 100644 --- a/examples/make_minimize_ex +++ b/examples/make_minimize_ex @@ -6,15 +6,13 @@ TARGET = minimize_ex all: minimize_ex -minimize_ex: minimize_ex.o minimize_unc.o newton.o momentum.o sgd.o bfgs.o broyd.o lmlsqr.o nlcgd.o finite_dif.o cyc_queue.o wait.o +minimize_ex: minimize_ex.o minimize_unc.o newton.o momentum.o sgd.o bfgs.o broyd.o lmlsqr.o nlcgd.o finite_dif.o cyc_queue.o line_min.o roots.o wolfe_step.o $(CXX) $(CXXFLAGS) -o $(TARGET) *.o $(LIBS) %.o: %.cpp $(DEPS) minimize_ex.o: examples/minimize_ex.cpp $(DEPS) $(CXX) -g -c examples/minimize_ex.cpp -wait.o: examples/wait.cpp - $(CXX) -g -c examples/wait.cpp clean: rm *.o \ No newline at end of file diff --git a/examples/minimize_ex.cpp b/examples/minimize_ex.cpp index 50b6ab3..ce44383 100644 --- a/examples/minimize_ex.cpp +++ b/examples/minimize_ex.cpp @@ -7,49 +7,69 @@ typedef std::vector vvd; void wait_for_key(); int main() { + int n = 10; // n=2 for rosenbrock, anything for the others arma::arma_rng::set_seed_random(); - arma::vec x = 5 - 10*arma::randu(100); - arma::vec w = arma::linspace(-5,5,x.n_elem); + + arma::vec x; + x = 5 - 10*arma::randu(n); // styb-tang or sphere + // x = {4*arma::randu()-2, 4*arma::randu() - 1}; // rosenbrock + if (n <= 5) std::cout << "initial point: " << x.t() << std::endl; + + arma::vec w = arma::linspace(-5,5,n); // for sphere + + arma::vec tru_min; + tru_min = arma::ones(n); + tru_min *= -2.9035; // for rosenbrock // sphere function // styblinski tang + // rosenbrock vec_dfunc g = [&w](const arma::vec& x) -> double { // return arma::norm(x - w); return arma::sum(x%x%x%x - 16*x%x + 5*x)/2; + // return std::pow(1-x(0),2) + 100*std::pow(x(1) - x(0)*x(0),2); }; vector_func dg = [&w](const arma::vec& x) -> arma::vec { // return 2*(x-w); return (2*x%x%x - 16*x + 2.5); + // return {-2*(1-x(0)) - 400*x(0)*(x(1)-x(0)*x(0)), + // 200*(x(1)-x(0)*x(0))}; }; sp_vector_func dgi = [&w](const arma::vec& x,int i) -> double { // return 2*(x(i) - w(i)); return (2*x(i)*x(i)*x(i) - 16*x(i) + 2.5); + // none for rosenbrock }; vec_mat_func J = [](const arma::vec& x) -> arma::mat { // return 2*arma::eye(x.n_elem, x.n_elem); return arma::diagmat(6*x%x - 16); + // return {{1200*x(0) - 400*x(1) + 2, -400*x(0)}, + // {-400*x(0), 200}}; }; optim_opts opts; // opts.use_bfgs(&dg); std::cout << "using BFGS..." << std::endl; // opts.use_momentum(&dg); std::cout << "using momentum gradient descent..." << std::endl; - // opts.use_sgd(&dgi); opts.stochastic_batch_size = 50; std::cout << "using batch stochastic gradient descent..." << std::endl; - // opts.use_lmlsqr(&dg); std::cout << "using Levenberg-Marquardt least squares..." << std::endl; + // opts.use_sgd(&dgi); opts.stochastic_batch_size = 5; std::cout << "using batch stochastic gradient descent..." << std::endl; + // opts.use_lmlsqr(&dg); opts.hessian_func = &J; std::cout << "using Levenberg-Marquardt least squares..." << std::endl; // opts.use_lbfgs(&dg); opts.num_iters_to_remember = 5; std::cout << "using limited memory BFGS..." << std::endl; - opts.use_lncgd(&dg); opts.hessian_func = &J; opts.tolerance = 1e-10; std::cout << "using conjugate gradient method..." << std::endl; + // opts.use_nlcgd(&dg); std::cout << "using conjugate gradient method..." << std::endl; // opts.use_newton(&dg,&J); std::cout << "using Newton's method..." << std::endl; - opts.use_FD_hessian = true; + opts.tolerance = 1e-5; + opts.max_iter = 1000; clock_t tt = clock(); - minimize_unc(g, x, opts); + double gx = numerics::minimize_unc(g, x, opts); tt = clock() - tt; - arma::vec centers = {-2.903,0,2.903}; + arma::vec centers; + centers = {-2.903,0.1567,2.746}; // styb-tang + // centers = {-2,-1,0,1,2}; // rosenbrock and sphere arma::uvec bins = arma::hist(x,centers); int i = arma::index_max(bins); - std::cout << "\noptimization results:\t\t" << g(x) << std::endl - << "true min:\t\t\t" << g(-2.903*arma::ones(x.n_elem)) << std::endl + std::cout << "\noptimization results:\t\t" << gx << std::endl + << "true min:\t\t\t" << g(tru_min) << std::endl << "mode(x):\t\t\t " << centers(i) << std::endl << "minimize_unc() took " << (float)tt/CLOCKS_PER_SEC << " seconds" << std::endl << "num iterations needed: " << opts.num_iters_returned << std::endl; diff --git a/examples/newton_ex.cpp b/examples/newton_ex.cpp index 00d0f99..bf34186 100644 --- a/examples/newton_ex.cpp +++ b/examples/newton_ex.cpp @@ -1,12 +1,14 @@ #include "../numerics.hpp" #include -// g++ -Wall -g -o newton_ex examples/newton_ex.cpp newton.cpp broyd.cpp lmlsqr.cpp finite_dif.cpp -larmadillo +// g++ -Wall -g -o newton_ex examples/newton_ex.cpp newton.cpp broyd.cpp lmlsqr.cpp finite_dif.cpp cyc_queue.cpp wolfe_step.cpp -larmadillo using namespace numerics; int main() { - auto f = [](const arma::vec& x) -> arma::vec { // function + arma::arma_rng::set_seed_random(); + + vector_func f = [](const arma::vec& x) -> arma::vec { // function arma::vec y = {0,0,0}; y(0) = x(0)*x(1) - 1; y(1) = x(0) + x(1)*x(2); @@ -26,11 +28,11 @@ int main() { arma::vec root = {-1.25992, -0.793701, -1.5874}; - std::cout << "In this file you can test the nonlinear solvers: Newton's method, BFGS, Broyden's method, and Levenberg-Marquardt." << std::endl; + std::cout << "In this file you can test the nonlinear solvers: Newton's method, Broyden's method, and Levenberg-Marquardt." << std::endl; std::cout << "we will try to find roots of f(x,y,z) = [xy - 1, x + yz, 2y - z] with initial guess [-1,-1,-1]." << std::endl; std::cout << "for newton we will need the jacobian: " << std::endl << "[y x 0]\n[1 z y]\n[0 2 -1]" << std::endl; std::cout << "we also know there is only one root at: [-1.25992, -0.793701, -1.5874]" << std::endl; - arma::vec x0 = {-1,-1,-1}; + arma::vec x0 = 4*arma::randu(3) - 2; nonlin_opts opts; opts.use_FD_jacobian = true; diff --git a/examples/roots_ex.cpp b/examples/roots_ex.cpp index d6c4c5e..cee2fbb 100644 --- a/examples/roots_ex.cpp +++ b/examples/roots_ex.cpp @@ -1,21 +1,20 @@ #include "../numerics.hpp" -// g++ -Wall -g -o roots_ex examples/roots_ex.cpp roots.cpp finite_dif.cpp +// g++ -Wall -g -o roots_ex examples/roots_ex.cpp roots.cpp using namespace numerics; -using std::sin; -using std::cos; +using std::exp; using std::pow; int main() { - auto g = [](double x) -> double { return sin(pow(x,2)); }; - auto dg = [](double x) -> double { return 2*x*cos(pow(x,2)); }; + auto g = [](double x) -> double { return exp(x) - pow(x,2); }; + auto dg = [](double x) -> double { return exp(x) - 2*x; }; double y; - std::cout << "let's try to find roots of: f(x) = sin[x^2]" << std::endl; + std::cout << "let's try to find roots of: f(x) = exp[x] - x^2" << std::endl; clock_t t = clock(); - y = newton(g, dg, 2); + y = newton(g, dg, 0); t = clock() - t; std::cout << "(1)\tfor newton's method we need the derivative of the function and a starting point:" << std::endl @@ -24,7 +23,7 @@ int main() { << "\t\tnewton() took: " << (float)t/CLOCKS_PER_SEC << " seconds" << std::endl << std::endl; t = clock(); - y = secant(g, 1.0); + y = secant(g, -3, 3); t = clock() - t; std::cout << "(2)\tfor secant method we need to starting points:" << std::endl @@ -33,7 +32,7 @@ int main() { << "\t\tsecant() took: " << (float)t/CLOCKS_PER_SEC << " seconds" << std::endl << std::endl; t = clock(); - y = bisect(g, 0.5, 2.0); + y = bisect(g, -3, 3); t = clock() - t; std::cout << "(3)\tfor bisection method we need to end points of a seach region:" << std::endl @@ -42,7 +41,7 @@ int main() { << "\t\tsecant() took: " << (float)t/CLOCKS_PER_SEC << " seconds" << std::endl << std::endl; t = clock(); - y = roots(g, 0.5, 2.0); + y = fzero(g, -3, 3); t = clock() - t; std::cout << "(4)\tfor roots() we need end points again but we adaptively use two methods to narrow our search region:" << std::endl diff --git a/finite_dif.cpp b/finite_dif.cpp index 29b3d01..26a727e 100755 --- a/finite_dif.cpp +++ b/finite_dif.cpp @@ -18,6 +18,13 @@ void numerics::approx_jacobian(const vector_func& f, arma::mat& J, const arma::v } } +arma::vec numerics::jacobian_diag(const vector_func& f, const arma::vec& x) { + double h = 1e-3; + arma::vec J = f(x - 2*h) - 8*f(x - h) + 8*f(x + h) - f(x + 2*h); + J /= 12*h; + return J; +} + //--- computes the gradient of a function of multiple variables ---// //----- f : f(x) whose gradient to approximate -------------------// //----- x : vector to evaluate gradient at -----------------------// diff --git a/fzero.cpp b/fzero.cpp new file mode 100755 index 0000000..4886e0e --- /dev/null +++ b/fzero.cpp @@ -0,0 +1,165 @@ +#include "numerics.hpp" + +//--- finds local root of single variable nonlinear functions using newton's method ---// +//----- f : function to find root of -------------------------------------------------// +//----- df : derivative of f ----------------------------------------------------------// +//----- x0 : point near the root ------------------------------------------------------// +double numerics::newton(const dfunc& f, const dfunc& df, double x, double err) { + err = std::abs(err); if (err <= 0) err = 1e-12; + int max_iter = 100; + + double s; + short k = 0; + do { + if (k >= max_iter) { // too many iterations + std::cerr << "newton() failed: too many iterations needed to converge." << std::endl + << "returing current best estimate." + << "!!!---not necessarily a good estimate---!!!" << std::endl + << "|f(x)| = " << std::abs(f(x)) << " > tolerance" << std::endl << std::endl; + return NAN; + } + s = -f(x)/df(x); + x += s; + k++; + } while (std::abs(s) > err); + return x; +} + +//--- secant methods for finding roots of single variable functions ---// +//--------- for added efficency we attempt ----------------------------// +//--------- to bracket the root with an auxilary point ----------------// +//----- f : function to find root of ---------------------------------// +//----- x0 : point near the root --------------------------------------// +double numerics::secant(const dfunc& f, double a, double b) { + double tol = 1e-8; int max_iter = 100; + + double fa = f(a), fb = f(b); + int k = 2; + if (std::abs(fa) < tol) return a; + if (std::abs(fb) < tol) return b; + + double c = (a+b)/2; + double fc = f(c); k++; + + while (std::abs(fc) > tol && std::abs(b-a) > tol) { + if (k >= max_iter) { // too many iterations + std::cerr << "secant() error: could not converge within " << max_iter << " function evaluations." << std::endl + << "\treturing current best estimate." + << "!!!---not necessarily a good estimate---!!!" << std::endl + << "|f(x)| = " << std::abs(fc) << " > tolerance" << std::endl << std::endl; + break; + } + double v; + + if (fc > 0) v = c - fc*(c-a)/(fc-fa); + else v = b - fb*(b-c)/(fb-fc); + + double fv = f(v); k++; + + if (fv < 0) { + a = c; + fa = fc; + } else { + b = c; + fb = fc; + } + c = v; + fc = fv; + } + return c; +} + +//--- bisection method for finding roots of single variable functions ---// +//----- f : function to find root of -----------------------------------// +//----- x1 : lower bound of search interval -----------------------------// +//----- x2 : upper bound of search interval -----------------------------// +double numerics::bisect(const dfunc& f, double a, double b, double tol) { + tol = std::abs(tol); int max_iter = 100; + short k = 2; + + double fa = f(a), fb = f(b); + + if (std::abs(fa) < tol ) { + return a; + } else if (std::abs(fb) < tol ) { + return b; + } + + if (fa*fb > 0) { // bracketing error + std::cerr << "bisect() error: provided points do not bracket a simple root." << std::endl; + return NAN; + } + + double c, fc; + + do { + if (k >= max_iter) { // too many iterations + std::cerr << "bisect() error: could not converge within " << max_iter << " function evaluations." << std::endl + << "returing current best estimate." + << "!!!---not necessarily a good estimate---!!!" << std::endl + << "|f(x)| = " << std::abs(fc) << " > tolerance" << std::endl << std::endl; + break; + } + c = (a+b)/2; + fc = f(c); k++; + if (std::abs(fc) < tol) break; + if (fc*fa < 0) { + b = c; + fb = fc; + } else { + a = c; + fa = fc; + } + } while (std::abs(fc) > tol && std::abs(b-a) > tol); + + return (a + b)/2; +} + +//--- narrows down to root using bisection method then converges using secant method ---// +//----- f : function to find root of --------------------------------------------------// +//----- x1 : lower bound of search interval --------------------------------------------// +//----- x2 : upper bound of search interval --------------------------------------------// +double numerics::fzero(const dfunc& f, double a, double b) { + double tol = 1e-10; int max_iter = 100; + + double fa = f(a), fb = f(b); + int k = 2; + if (std::abs(fa) < tol) return a; + if (std::abs(fb) < tol) return b; + + if (fa*fb > 0) { + std::cerr << "fzero() error: provided points do not bracket a simple root." << std::endl; + return NAN; + } + + double c = (a+b)/2; + double fc = f(c); k++; + + while (std::abs(fc) > tol && std::abs(b-a) > tol) { + if (k >= max_iter) { // error + std::cerr << "fzero() error: could not converge within " << max_iter << " function evaluations." << std::endl + << "returing current best estimate." + << "!!!---not necessarily a good estimate---!!!" << std::endl + << "|f(x)| = " << std::abs(fc) << " > tolerance" << std::endl << std::endl; + break; + } + double v; + if (std::abs(fa-fc) > tol || std::abs(fb-fc) > tol) { // inverse quadratic + v = (a*fb*fc)/((fa-fb)*(fa-fc)) + (b*fa*fc)/((fb-fa)*(fb-fc)) + (c*fa*fb)/((fc-fa)*(fc-fb)); + } else { // secant + if (fc > 0) v = c - fc*(c-a)/(fc-fa); + else v = b - fb*(b-c)/(fb-fc); + } + double fv = f(v); k++; + if (fv < 0) { + a = c; + fa = fc; + } else { + b = c; + fb = fc; + } + c = v; + fc = fv; + } + return c; +} \ No newline at end of file diff --git a/line_min.cpp b/line_min.cpp new file mode 100644 index 0000000..6a6c747 --- /dev/null +++ b/line_min.cpp @@ -0,0 +1,66 @@ +#include "numerics.hpp" + +double numerics::line_min(const dfunc& f) { + int max_iter_i = 100; + int max_iter = 100; + double tol = 1e-8; + double interp_tol = 1e-8; + + double a = 0, + b = 1, + s = 0.75, + t = 1.25, + fb = f(b), + c, fa, fc, v; + int k = 1; + + // -- initialise bracket + while (fb < 0) { + if (k >= max_iter_i) break; + double fb1 = f(s*b); + if (fb1 < 0) { // s*b does not bracket a minimum + double fb2 = f(t*b); + if (fb2 > 0 || fb2 > fb1) { // t*b does brack a minimum or is closer to a min + b *= t; + fb = fb2; + std::swap(s,t); + } else { // s*b is closer than t*b to the minimum + b *= s; + fb = fb1; + } + } else { + b *= s; + fb = fb1; + } + k++; + } + fa = f(a); + + // -- begin root finding over bracket + if (std::abs(fb) < tol) return b; + c = (a+b)/2; + fc = f(c); + + k = 1; + while (std::abs(fc) > tol && std::abs(b-a) > tol) { + if (k >= max_iter) break; + if (std::abs(fa-fc) > interp_tol && std::abs(fb-fc) > interp_tol) { // inverse quadratic + v = (a*fb*fc)/((fa-fb)*(fa-fc)) + (b*fa*fc)/((fb-fa)*(fb-fc)) + (c*fa*fb)/((fc-fa)*(fc-fb)); + } else { // secant + if (fc > 0) v = c - fc*(c-a)/(fc-fa); + else v = b - fb*(b-c)/(fb-fc); + } + double fv = f(v); + if (fv < 0) { + a = c; + fa = fc; + } else { + b = c; + fb = fc; + } + c = v; + fc = fv; + k++; + } + return c; +} \ No newline at end of file diff --git a/minimize_unc.cpp b/minimize_unc.cpp index beb85bf..79e6cda 100644 --- a/minimize_unc.cpp +++ b/minimize_unc.cpp @@ -22,14 +22,14 @@ double numerics::minimize_unc(const vec_dfunc& f, arma::vec& x, optim_opts& opts << "\talternatively enabling FD hessian evaluations can be efficient for well conditioned objective functions." << std::endl << "3) L-BFGS (or limited memory BFGS) has the benefits of BFGS but is more memory efficient." << std::endl << "\tif memory is a problem then L-BFGS should be used over BFGS, though BFGS typically converges slightly faster." << std::endl - << "4) Levenberg-Marquardt behaves similarly to BFGS and is the prefered method for nonlinear least squares problems," << std::endl + << "4) The nonlinear conjugate gradient method is optimal when optimizing in few variables, and when hessian computations are expensive" << std::endl + << "5) Levenberg-Marquardt behaves similarly to BFGS and is the prefered method for nonlinear least squares problems," << std::endl << "\tif you are solving a least squares problem, consider instead using lmlsqr() and providing the system g_i(b) = y_i - f(x_i;b)" << std::endl - << "5) Momentum gradient descent is used when Hessian based computations are too expensive," << std::endl + << "6) Momentum gradient descent is used when Hessian based computations are too expensive," << std::endl << "\tthis method typically requires many more iterations than Hessian based methods" << std::endl - << "6) Stochastic gradient descent is used when even gradient based computations are too expensive." << std::endl - << "\tchoosing a batch size, max number of iterations, and momentum parameters are all important choices for the success of this algorithm" << std::endl - << "7) Broyden's method provides an alternative as well, but BFGS is the prefered alternative for optimization." << std::endl - << "\tBroyden's method is ideal for generic root finding; for such problems, consider using broyd() instead." << std::endl + << "7) Stochastic gradient descent is used when even gradient based computations are too expensive." << std::endl + << "\tchoosing a batch size and max number of iterations are all important choices for the success of this algorithm" << std::endl + << "\tFor generic root finding use Broyden's method provided by broyd()." << std::endl << std::endl << "\t\t\tGood Luck!" << std::endl; } @@ -38,7 +38,6 @@ double numerics::minimize_unc(const vec_dfunc& f, arma::vec& x, optim_opts& opts options.err = opts.tolerance; options.max_iter = opts.max_iter; options.damping_param = opts.damping_param; - options.step_size = opts.step_size; if (opts.gradient_func == nullptr) { // no gradient provided std::cerr << "minimize_unc() warning: using FD gradients may be very slow!" << std::endl; options.max_iter = no_grad_max_iter; @@ -155,8 +154,6 @@ double numerics::minimize_unc(const vec_dfunc& f, arma::vec& x, optim_opts& opts gd_opts options; options.err = opts.tolerance; options.max_iter = opts.max_iter; - options.damping_param = opts.damping_param; - options.step_size = opts.step_size; options.stochastic_batch_size = opts.stochastic_batch_size; if (opts.indexed_gradient_func == nullptr) { // no gradient provided std::cerr << "minimize_unc() warning: using FD gradients may be very slow!" << std::endl; @@ -189,6 +186,7 @@ double numerics::minimize_unc(const vec_dfunc& f, arma::vec& x, optim_opts& opts } else { nlcgd(*opts.gradient_func, x, options); } + opts.num_iters_returned = options.num_iters_returned; } return f(x); diff --git a/momentum.cpp b/momentum.cpp index 21d157f..9f4bdf2 100644 --- a/momentum.cpp +++ b/momentum.cpp @@ -5,14 +5,16 @@ //----- x : initial guess ---------// //----- opts: parameter options ---// void numerics::mgd(const vector_func& f, arma::vec& x, gd_opts& opts) { - double gamma = opts.damping_param; - double alpha = opts.step_size; + double beta = opts.damping_param; arma::vec p = f(x); + double r = arma::norm(p,"inf"); + double alpha = line_min( [&p,&x,&f,r](double a) -> double {arma::vec q = (-1.0/r)*p; return arma::dot(q,f(x + a*q));} ); + x += (-alpha/r)*p; - size_t k = 0; - while (arma::norm(p,"inf") > opts.err) { - if (k > opts.max_iter) { // too many iterations + size_t k = 1; + while (std::abs(alpha*r) > opts.err) { + if (k >= opts.max_iter) { // too many iterations std::cerr << "mgd() error: too many iterations needed for convegence to a root." << std::endl << "returning current best result." << std::endl << "!!!---not necessarily a good estimate---!!!" << std::endl @@ -20,8 +22,10 @@ void numerics::mgd(const vector_func& f, arma::vec& x, gd_opts& opts) { opts.num_iters_returned = k; return; } - x -= alpha*p; - p = gamma*p + f(x); + p = beta*p + f(x); + r = arma::norm(p,"inf"); + alpha = line_min( [&p,&x,&f,r](double a) -> double {arma::vec q = (-1.0/r)*p; return arma::dot(q,f(x + a*q));} ); + x += (-alpha/r)*p; k++; } opts.num_iters_returned = k; diff --git a/nlcgd.cpp b/nlcgd.cpp index bf9a92a..af23e3e 100644 --- a/nlcgd.cpp +++ b/nlcgd.cpp @@ -1,72 +1,40 @@ #include "numerics.hpp" -//--- trust region based nonlinear conjugate gradient method ---// -//----- f : gradient function ----------------------------------// -//----- x : guess, and solution --------------------------------// +//--- nonlinear conjugate gradient method ---// +//----- f : gradient function ---------------// +//----- x : guess, and solution -------------// void numerics::nlcgd(const vector_func& f, arma::vec& x, nonlin_opts& opts) { int n = x.n_elem; - arma::vec p = -f(x); - arma::vec s = p; - - arma::mat I = arma::eye(n,n); - arma::mat B; - if (opts.init_jacobian == nullptr) { // no initial jacobian - if (opts.jacobian_func != nullptr) { // jacobian function provided - B = opts.jacobian_func->operator()(x); - } else if (opts.use_FD_jacobian) { // FD jacobian - approx_jacobian(f,B,x); - } else { // identity - B = arma::eye(n,n); - } - } else { // initial jacobian - B = *opts.init_jacobian; - } + arma::vec p; + arma::vec prev_x; + double alpha; + double r; size_t k = 0; - while (arma::norm(s,"inf") > opts.err) { - if (k >= opts.max_iter) { + do { + if (k >= opts.max_iter) { // too many iterations std::cerr << "\nnlcgd() failed: too many iterations needed for convergence." << std::endl << "returning current best estimate." << std::endl << "!!!---not necessarily a good estimate---!!!" << std::endl - << "||f(x)|| = " << arma::norm(f(x), "inf") << " > 0" << std::endl << std::endl; + << "||f(x)|| = " << arma::norm(p, "inf") << " > 0" << std::endl << std::endl; break; } - arma::vec fx = f(x); - double alpha = arma::dot(fx,fx) / arma::dot(p, B*p); - double sts = arma::dot(s,s); - arma::vec ds = alpha*p - s; - - s = alpha*p; - x += s; - k++; - - arma::vec fx1 = -f(x); - if (mod(k,n) == 0) { - p = fx1; - if (opts.jacobian_func == nullptr) { // BFGS update - arma::vec y = -(fx1 + fx); - double sy = arma::dot(s,y); - arma::vec BS = B*s; - B += y*y.t()/sy - BS*BS.t()/arma::dot(s,BS); - } else { // Hessian provided - B = opts.jacobian_func->operator()(x); - } - } else { - double beta = std::max(arma::dot(s,ds)/sts, 0.0); - p = fx1 + beta*p; + if (2*n-4 == 0) prev_x = x; + for (int i(1); i <= 2*n-2; ++i) { + p = -f(x); + r = arma::norm(p,"inf"); + alpha = line_min( [&p,&x,&f,r](double a) -> double {return arma::dot(p/r,f(x + (a/r)*p));} ); + x += (alpha/r)*p; + k++; + if (i == 2*n - 4) prev_x = x; } - - // if (opts.jacobian_func == nullptr) { // BFGS update - // arma::vec y = -(fx1 + fx); - // double sy = arma::dot(s,y); - // arma::vec BS = B*s; - // B += y*y.t()/sy - BS*BS.t()/arma::dot(s,BS); - // } else { // Hessian provided - // B = opts.jacobian_func->operator()(x); - // } - - } + p = prev_x - x; + r = arma::norm(p,"inf"); + alpha = line_min( [&p,&x,&f,r](double a) -> double {return arma::dot(p/r,f(x + (a/r)*p));} ); + x += (alpha/r)*p; + k++; + } while (std::abs(alpha*r) > opts.err); opts.num_iters_returned = k; } diff --git a/numerics.hpp b/numerics.hpp index 451eb04..946c905 100755 --- a/numerics.hpp +++ b/numerics.hpp @@ -9,6 +9,20 @@ #define ARMA_USE_SUPERLU 1 #include +/* Copyright 2019 Amit Rotem + +Licensed under the Apache License, Version 2.0 (the "License"); +you may not use this file except in compliance with the License. +You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + +Unless required by applicable law or agreed to in writing, software +distributed under the License is distributed on an "AS IS" BASIS, +WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +See the License for the specific language governing permissions and +limitations under the License. */ + namespace numerics { // --- utitility -------------- // // --- integral constants @@ -155,7 +169,6 @@ namespace numerics { size_t max_iter; size_t grad_nelem; double damping_param; - double step_size; size_t stochastic_batch_size; // outputs @@ -165,7 +178,6 @@ namespace numerics { max_iter = gd_max_iter; grad_nelem = 0; damping_param = 0.99; - step_size = 3e-3; num_iters_returned = 0; stochastic_batch_size = 10; } @@ -184,7 +196,6 @@ namespace numerics { double wolfe_c1; // -- lbfgs, bfgs double wolfe_c2; // -- lbfgs, bfgs double wolfe_scaling; // -- lbfgs, bfgs - double step_size; // -- mgd, sgd size_t stochastic_batch_size; // -- sgd size_t num_iters_to_remember; // -- lbfgs arma::mat* init_hessian; // -- bfgs, lbfgs, lmlsqr @@ -207,7 +218,6 @@ namespace numerics { wolfe_c1 = 0; wolfe_c2 = 0; wolfe_scaling = 0; - step_size = 3e-3; init_hessian = nullptr; init_hessian_inv = nullptr; hessian_func = nullptr; @@ -237,7 +247,6 @@ namespace numerics { wolfe_c1 = 1e-4; wolfe_c2 = 0.9; wolfe_scaling = 0.5; - step_size = 0; init_hessian = nullptr; init_hessian_inv = nullptr; stochastic_batch_size = 0; @@ -262,7 +271,6 @@ namespace numerics { wolfe_c1 = 1e-4; wolfe_c2 = 0.9; wolfe_scaling = 0.5; - step_size = 0; init_hessian = nullptr; init_hessian_inv = nullptr; stochastic_batch_size = 0; @@ -288,7 +296,6 @@ namespace numerics { wolfe_c1 = 1e-4; wolfe_c2 = 0.9; wolfe_scaling = 0.5; - step_size = 0; init_hessian = nullptr; init_hessian_inv = nullptr; stochastic_batch_size = 0; @@ -314,7 +321,6 @@ namespace numerics { wolfe_c1 = 0; wolfe_c2 = 0; wolfe_scaling = 0; - step_size = 0; init_hessian = nullptr; init_hessian_inv = nullptr; stochastic_batch_size = 0; @@ -340,7 +346,6 @@ namespace numerics { wolfe_c1 = 0; wolfe_c2 = 0; wolfe_scaling = 0; - step_size = 3e-3; init_hessian = nullptr; init_hessian_inv = nullptr; stochastic_batch_size = 0; @@ -366,15 +371,35 @@ namespace numerics { wolfe_c1 = 0; wolfe_c2 = 0; wolfe_scaling = 0; - step_size = 5e-2; init_hessian = nullptr; init_hessian_inv = nullptr; stochastic_batch_size = 10; num_iters_to_remember = 0; } - void use_lncgd(vector_func* gradient) { - use_bfgs(gradient); + void use_nlcgd(vector_func* gradient) { + if (gradient == nullptr) { + std::cerr << "optim_opts::use_nlcgd() error: nlcgd requires a gradient input." << std::endl; + return; + } + solver = NLCGD; + tolerance = root_err; + max_iter = bfgs_max_iter; + hessian_func = nullptr; + gradient_func = gradient; + indexed_gradient_func = nullptr; + use_FD_gradient = false; + use_FD_hessian = false; + use_scale_invariance = false; + damping_param = 0; + damping_scale = 0; + wolfe_c1 = 0; + wolfe_c2 = 0; + wolfe_scaling = 0; + init_hessian = nullptr; + init_hessian_inv = nullptr; + stochastic_batch_size = 0; + num_iters_to_remember = 0; } } optim_opts; @@ -416,17 +441,21 @@ namespace numerics { size_t max_elem; size_t size; size_t head; - arma::mat A; public: + arma::mat A; cyc_queue(size_t num_rows, size_t max_size); - void push(arma::vec& x); + void push(const arma::vec& x); arma::vec operator()(size_t i); + arma::vec end(); int length(); int col_size(); + void clear(); + arma::mat data(); }; double wolfe_step(const vec_dfunc&, const vector_func&, const arma::vec&, const arma::vec&, double, double, double); + double line_min(const dfunc&); // --- integration ------------ // double integrate(const dfunc&, double, double, integrator i = SIMPSON, double err = 1e-5); double Sintegrate(const dfunc&, double, double, double err = 1e-5); @@ -469,10 +498,10 @@ namespace numerics { gd_opts sgd(const sp_vector_func&, arma::vec&); //--- univariate ---// + double fzero(const dfunc&, double, double); double newton(const dfunc&, const dfunc&, double, double err = 1e-10); - double secant(const dfunc&, double, double err = 1e-8); + double secant(const dfunc&, double, double err = 1e-10); double bisect(const dfunc&, double, double, double tol = 1e-8); - double roots(const dfunc&, double, double); // --- optimization ----------- // double minimize_unc(const vec_dfunc&, arma::vec&, optim_opts&); double minimize_unc(const vec_dfunc&, arma::vec&); @@ -525,6 +554,8 @@ namespace numerics { arma::mat lagrangeInterp(const arma::vec&, const arma::mat&, const arma::vec&); arma::mat sincInterp(const arma::vec&, const arma::mat&, const arma::vec&); // --- difference methods ----- // + arma::vec jacobian_diag(const vector_func&, const arma::vec&); + void approx_jacobian(const vector_func&, arma::mat&, const arma::vec&, double err = 1e-2, bool catch_zero = true); arma::vec grad(const vec_dfunc&, const arma::vec&, double err = 1e-5, bool catch_zero = true); diff --git a/pois_ex b/pois_ex deleted file mode 100755 index 6aae46ec6de9ee77133ffa0d88fc55780673d1e9..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2328224 zcmeFa33yf26*hbUqd~z06-^y#9I$c1phSa+#vyu9(8!>mXu=R6$`B#JIG~~7TCTU( zrnYIRl{R%~u~kc3(P#w&B0+20sIA6UOX^V1HMFKy+c@&S?^^quv(LHrGT46q_x#WE zhbNq~-o5tPYp=ETI(zRk4)>~(-bH(c-7LQ%vmH-7AcJk z#Q(!0M@RMqx(9yCOlkPoM`8GJkGV=l*dXhb_E#W$Jk;>HM)f z7>N`L&HB`JWt*0-Y&%+T_qbv&AtU*N_2|~4daPIUgw`v1LhHH5Vxzx%bo)nc{!TLG z;p2HICTtP+=+?Uw^{D5r{^T3R38uYlcA0DJ;vQW; 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-//----- df : derivative of f ----------------------------------------------------------// -//----- x0 : point near the root ------------------------------------------------------// -double numerics::newton(const dfunc& f, const dfunc& df, double x0, double err) { - auto p = [&f,&df](double x) -> double { return x - f(x)/df(x); }; - double x1 = x0; - double x2 = p(x1); - short k = 0; - while (std::abs(x2-x1) > err) { - if (k > 100) { - std::cerr << "newton() failed: too many iterations needed to converge." << std::endl; - return NAN; - } - x1 = x2; - x2 = p(x2); - k++; - } - return x2; -} - -//--- secant methods for finding roots of single variable functions ---// -//----- f : function to find root of ---------------------------------// -//----- x0 : point near the root --------------------------------------// -double numerics::secant(const dfunc& f, double x0, double err) { - auto df = [&](double x) -> double { - return deriv(f,x,err); - }; - return newton(f,df,x0,err); -} - -//--- bisection method for finding roots of single variable functions ---// -//----- f : function to find root of -----------------------------------// -//----- x1 : lower bound of search interval -----------------------------// -//----- x2 : upper bound of search interval -----------------------------// -double numerics::bisect(const dfunc& f, double x1, double x2, double tol) { - if (tol <= 0) tol = eps(x2); // check tolerance condition - if (std::abs(f(x1)) < tol ) { - return x1; - } else if (std::abs(f(x2)) < tol ) { - return x2; - } - - while ( std::abs(x2 - x1) > tol ) { - double p = (x2 + x1)/2; - if ( std::abs(f(p)) < tol ) { - return p; - } else if (f(x1) * f(p) < 0) { - x2 = p; - } else { - x1 = p; - } - } - - if ( std::abs(f(x2)) > tol ) { // error control - std::cerr << "bisect() failed: there is likely no simple roots on the interval." << std::endl; - return NAN; - } - return (x1 + x2)/2; -} - - -//--- narrows down to root using bisection method then converges using secant method ---// -//----- f : function to find root of --------------------------------------------------// -//----- x1 : lower bound of search interval --------------------------------------------// -//----- x2 : upper bound of search interval --------------------------------------------// -double numerics::roots(const dfunc& f, double x1, double x2) { - if (std::abs(f(x1)) < eps(x1)) { - return x1; - } else if (std::abs(f(x2)) < eps(x2)) { - return x2; - } - - // --- bisect to reduce problem - double p; - while (std::abs(x2 - x1) > 0.1) { - p = (x2 + x1)/2; - if ( std::abs(f(p)) < eps(p) ) { - return p; - } else if (f(x1) * f(p) < 0) { - x2 = p; - } else { - x1 = p; - } - } - - // --- secant to the rest of the way - p = (x2 + x1)/2; - return secant(f,p,1e-10); -} \ No newline at end of file diff --git a/sgd.cpp b/sgd.cpp index c544095..f5081a5 100644 --- a/sgd.cpp +++ b/sgd.cpp @@ -7,18 +7,13 @@ void numerics::sgd(const sp_vector_func& f, arma::vec& x, gd_opts& opts) { int n = x.n_elem; int bs = opts.stochastic_batch_size; - double alpha = opts.step_size; - double gamma = opts.damping_param; - arma::vec p = arma::zeros(n); - arma::uvec ind = arma::shuffle( arma::regspace(0,n-1) ); - ind = ind( arma::span(0,bs-1) ); - for (int i(0); i < bs; ++i) { - p( ind(i) ) = f( x, ind(i) ); - } + double alpha; + double r; + arma::vec p; size_t k = 0; - while (arma::norm(p,"inf") > opts.err) { + do { if (k > opts.max_iter) { // too many iterations std::cerr << "sgd() error: too many iterations needed for convegence to a root." << std::endl << "returning current best result." << std::endl @@ -27,17 +22,26 @@ void numerics::sgd(const sp_vector_func& f, arma::vec& x, gd_opts& opts) { opts.num_iters_returned = k; return; } - x -= alpha*p; - - arma::vec g = arma::zeros(n); - ind = arma::shuffle( arma::regspace(0,n-1) ); + p = arma::zeros(n); + arma::uvec ind = arma::shuffle( arma::regspace(0,n-1) ); ind = ind( arma::span(0,bs-1) ); for (int i(0); i < bs; ++i) { - g( ind(i) ) = f( x, ind(i) ); + p( ind(i) ) = f( x, ind(i) ); } - p = gamma*p + g; + r = arma::norm(p,"inf"); + alpha = line_min( + [&f,&ind,&p,&x,n,bs,r](double a) -> double { + arma::vec z = arma::zeros(n); + arma::vec q = (-1.0/r)*p; + for (int i(0); i < bs; ++i) { + z( ind(i) ) = f(x + a*q, ind(i)); + } + return arma::dot(q,z); + } + ); + x += (-alpha/r)*p; k++; - } + } while (std::abs(alpha*r) > opts.err); opts.num_iters_returned = k; } diff --git a/statistics/statistics.hpp b/statistics/statistics.hpp index f2da1f9..2b452ef 100644 --- a/statistics/statistics.hpp +++ b/statistics/statistics.hpp @@ -5,6 +5,20 @@ using namespace arma; +/* Copyright 2019 Amit Rotem + +Licensed under the Apache License, Version 2.0 (the "License"); +you may not use this file except in compliance with the License. +You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + +Unless required by applicable law or agreed to in writing, software +distributed under the License is distributed on an "AS IS" BASIS, +WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +See the License for the specific language governing permissions and +limitations under the License. */ + namespace statistics { enum hypothesis { NEQ, diff --git a/wolfe_step.cpp b/wolfe_step.cpp new file mode 100644 index 0000000..0c0e043 --- /dev/null +++ b/wolfe_step.cpp @@ -0,0 +1,18 @@ +#include "numerics.hpp" + +//--- rough approximator step size for quasi-newton methods based on strong wolfe conditions ---// +double numerics::wolfe_step(const vec_dfunc& obj_func, const vector_func& f, const arma::vec& x, const arma::vec& p, double c1, double c2, double b) { + double alpha = 1; + int k = 0; + while (true) { + if (k > 100) break; + double pfa = arma::dot(p, f(x + alpha*x)); + bool cond1 = obj_func(x + alpha*p) <= obj_func(x) + c1*alpha*pfa; + bool cond2 = std::abs(arma::dot(p, f(x + alpha*p))) <= c2*std::abs(arma::dot(p, f(x))); + if (cond1 && cond2) break; + else if (obj_func(x + b*alpha*p) < obj_func(x + alpha*p)) alpha *= b; + else alpha /= b; + k++; + } + return alpha; +} \ No newline at end of file