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diff --git a/training/lbfgs.h b/training/lbfgs.h new file mode 100644 index 00000000..e8baecab --- /dev/null +++ b/training/lbfgs.h @@ -0,0 +1,1459 @@ +#ifndef SCITBX_LBFGS_H +#define SCITBX_LBFGS_H + +#include <cstdio> +#include <cstddef> +#include <cmath> +#include <stdexcept> +#include <algorithm> +#include <vector> +#include <string> +#include <iostream> +#include <sstream> + +namespace scitbx { + +//! Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) %minimizer. +/*! Implementation of the + Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) + algorithm for large-scale multidimensional minimization + problems. + + This code was manually derived from Java code which was + in turn derived from the Fortran program + <code>lbfgs.f</code>. The Java translation was + effected mostly mechanically, with some manual + clean-up; in particular, array indices start at 0 + instead of 1. Most of the comments from the Fortran + code have been pasted in. + + Information on the original LBFGS Fortran source code is + available at + http://www.netlib.org/opt/lbfgs_um.shar . The following + information is taken verbatim from the Netlib documentation + for the Fortran source. + + <pre> + file opt/lbfgs_um.shar + for unconstrained optimization problems + alg limited memory BFGS method + by J. Nocedal + contact nocedal@eecs.nwu.edu + ref D. C. Liu and J. Nocedal, ``On the limited memory BFGS method for + , large scale optimization methods'' Mathematical Programming 45 + , (1989), pp. 503-528. + , (Postscript file of this paper is available via anonymous ftp + , to eecs.nwu.edu in the directory pub/%lbfgs/lbfgs_um.) + </pre> + + @author Jorge Nocedal: original Fortran version, including comments + (July 1990).<br> + Robert Dodier: Java translation, August 1997.<br> + Ralf W. Grosse-Kunstleve: C++ port, March 2002.<br> + Chris Dyer: serialize/deserialize functionality + */ +namespace lbfgs { + + //! Generic exception class for %lbfgs %error messages. + /*! All exceptions thrown by the minimizer are derived from this class. + */ + class error : public std::exception { + public: + //! Constructor. + error(std::string const& msg) throw() + : msg_("lbfgs error: " + msg) + {} + //! Access to error message. + virtual const char* what() const throw() { return msg_.c_str(); } + protected: + virtual ~error() throw() {} + std::string msg_; + public: + static std::string itoa(unsigned long i) { + std::ostringstream os; + os << i; + return os.str(); + } + }; + + //! Specific exception class. + class error_internal_error : public error { + public: + //! Constructor. + error_internal_error(const char* file, unsigned long line) throw() + : error( + "Internal Error: " + std::string(file) + "(" + itoa(line) + ")") + {} + }; + + //! Specific exception class. + class error_improper_input_parameter : public error { + public: + //! Constructor. + error_improper_input_parameter(std::string const& msg) throw() + : error("Improper input parameter: " + msg) + {} + }; + + //! Specific exception class. + class error_improper_input_data : public error { + public: + //! Constructor. + error_improper_input_data(std::string const& msg) throw() + : error("Improper input data: " + msg) + {} + }; + + //! Specific exception class. + class error_search_direction_not_descent : public error { + public: + //! Constructor. + error_search_direction_not_descent() throw() + : error("The search direction is not a descent direction.") + {} + }; + + //! Specific exception class. + class error_line_search_failed : public error { + public: + //! Constructor. + error_line_search_failed(std::string const& msg) throw() + : error("Line search failed: " + msg) + {} + }; + + //! Specific exception class. + class error_line_search_failed_rounding_errors + : public error_line_search_failed { + public: + //! Constructor. + error_line_search_failed_rounding_errors(std::string const& msg) throw() + : error_line_search_failed(msg) + {} + }; + + namespace detail { + + template <typename NumType> + inline + NumType + pow2(NumType const& x) { return x * x; } + + template <typename NumType> + inline + NumType + abs(NumType const& x) { + if (x < NumType(0)) return -x; + return x; + } + + // This class implements an algorithm for multi-dimensional line search. + template <typename FloatType, typename SizeType = std::size_t> + class mcsrch + { + protected: + int infoc; + FloatType dginit; + bool brackt; + bool stage1; + FloatType finit; + FloatType dgtest; + FloatType width; + FloatType width1; + FloatType stx; + FloatType fx; + FloatType dgx; + FloatType sty; + FloatType fy; + FloatType dgy; + FloatType stmin; + FloatType stmax; + + static FloatType const& max3( + FloatType const& x, + FloatType const& y, + FloatType const& z) + { + return x < y ? (y < z ? z : y ) : (x < z ? z : x ); + } + + public: + /* Minimize a function along a search direction. This code is + a Java translation of the function <code>MCSRCH</code> from + <code>lbfgs.f</code>, which in turn is a slight modification + of the subroutine <code>CSRCH</code> of More' and Thuente. + The changes are to allow reverse communication, and do not + affect the performance of the routine. This function, in turn, + calls <code>mcstep</code>.<p> + + The Java translation was effected mostly mechanically, with + some manual clean-up; in particular, array indices start at 0 + instead of 1. Most of the comments from the Fortran code have + been pasted in here as well.<p> + + The purpose of <code>mcsrch</code> is to find a step which + satisfies a sufficient decrease condition and a curvature + condition.<p> + + At each stage this function updates an interval of uncertainty + with endpoints <code>stx</code> and <code>sty</code>. The + interval of uncertainty is initially chosen so that it + contains a minimizer of the modified function + <pre> + f(x+stp*s) - f(x) - ftol*stp*(gradf(x)'s). + </pre> + If a step is obtained for which the modified function has a + nonpositive function value and nonnegative derivative, then + the interval of uncertainty is chosen so that it contains a + minimizer of <code>f(x+stp*s)</code>.<p> + + The algorithm is designed to find a step which satisfies + the sufficient decrease condition + <pre> + f(x+stp*s) <= f(X) + ftol*stp*(gradf(x)'s), + </pre> + and the curvature condition + <pre> + abs(gradf(x+stp*s)'s)) <= gtol*abs(gradf(x)'s). + </pre> + If <code>ftol</code> is less than <code>gtol</code> and if, + for example, the function is bounded below, then there is + always a step which satisfies both conditions. If no step can + be found which satisfies both conditions, then the algorithm + usually stops when rounding errors prevent further progress. + In this case <code>stp</code> only satisfies the sufficient + decrease condition.<p> + + @author Original Fortran version by Jorge J. More' and + David J. Thuente as part of the Minpack project, June 1983, + Argonne National Laboratory. Java translation by Robert + Dodier, August 1997. + + @param n The number of variables. + + @param x On entry this contains the base point for the line + search. On exit it contains <code>x + stp*s</code>. + + @param f On entry this contains the value of the objective + function at <code>x</code>. On exit it contains the value + of the objective function at <code>x + stp*s</code>. + + @param g On entry this contains the gradient of the objective + function at <code>x</code>. On exit it contains the gradient + at <code>x + stp*s</code>. + + @param s The search direction. + + @param stp On entry this contains an initial estimate of a + satifactory step length. On exit <code>stp</code> contains + the final estimate. + + @param ftol Tolerance for the sufficient decrease condition. + + @param xtol Termination occurs when the relative width of the + interval of uncertainty is at most <code>xtol</code>. + + @param maxfev Termination occurs when the number of evaluations + of the objective function is at least <code>maxfev</code> by + the end of an iteration. + + @param info This is an output variable, which can have these + values: + <ul> + <li><code>info = -1</code> A return is made to compute + the function and gradient. + <li><code>info = 1</code> The sufficient decrease condition + and the directional derivative condition hold. + </ul> + + @param nfev On exit, this is set to the number of function + evaluations. + + @param wa Temporary storage array, of length <code>n</code>. + */ + void run( + FloatType const& gtol, + FloatType const& stpmin, + FloatType const& stpmax, + SizeType n, + FloatType* x, + FloatType f, + const FloatType* g, + FloatType* s, + SizeType is0, + FloatType& stp, + FloatType ftol, + FloatType xtol, + SizeType maxfev, + int& info, + SizeType& nfev, + FloatType* wa); + + /* The purpose of this function is to compute a safeguarded step + for a linesearch and to update an interval of uncertainty for + a minimizer of the function.<p> + + The parameter <code>stx</code> contains the step with the + least function value. The parameter <code>stp</code> contains + the current step. It is assumed that the derivative at + <code>stx</code> is negative in the direction of the step. If + <code>brackt</code> is <code>true</code> when + <code>mcstep</code> returns then a minimizer has been + bracketed in an interval of uncertainty with endpoints + <code>stx</code> and <code>sty</code>.<p> + + Variables that must be modified by <code>mcstep</code> are + implemented as 1-element arrays. + + @param stx Step at the best step obtained so far. + This variable is modified by <code>mcstep</code>. + @param fx Function value at the best step obtained so far. + This variable is modified by <code>mcstep</code>. + @param dx Derivative at the best step obtained so far. + The derivative must be negative in the direction of the + step, that is, <code>dx</code> and <code>stp-stx</code> must + have opposite signs. This variable is modified by + <code>mcstep</code>. + + @param sty Step at the other endpoint of the interval of + uncertainty. This variable is modified by <code>mcstep</code>. + @param fy Function value at the other endpoint of the interval + of uncertainty. This variable is modified by + <code>mcstep</code>. + + @param dy Derivative at the other endpoint of the interval of + uncertainty. This variable is modified by <code>mcstep</code>. + + @param stp Step at the current step. If <code>brackt</code> is set + then on input <code>stp</code> must be between <code>stx</code> + and <code>sty</code>. On output <code>stp</code> is set to the + new step. + @param fp Function value at the current step. + @param dp Derivative at the current step. + + @param brackt Tells whether a minimizer has been bracketed. + If the minimizer has not been bracketed, then on input this + variable must be set <code>false</code>. If the minimizer has + been bracketed, then on output this variable is + <code>true</code>. + + @param stpmin Lower bound for the step. + @param stpmax Upper bound for the step. + + If the return value is 1, 2, 3, or 4, then the step has + been computed successfully. A return value of 0 indicates + improper input parameters. + + @author Jorge J. More, David J. Thuente: original Fortran version, + as part of Minpack project. Argonne Nat'l Laboratory, June 1983. + Robert Dodier: Java translation, August 1997. + */ + static int mcstep( + FloatType& stx, + FloatType& fx, + FloatType& dx, + FloatType& sty, + FloatType& fy, + FloatType& dy, + FloatType& stp, + FloatType fp, + FloatType dp, + bool& brackt, + FloatType stpmin, + FloatType stpmax); + + void serialize(std::ostream* out) const { + out->write((const char*)&infoc,sizeof(infoc)); + out->write((const char*)&dginit,sizeof(dginit)); + out->write((const char*)&brackt,sizeof(brackt)); + out->write((const char*)&stage1,sizeof(stage1)); + out->write((const char*)&finit,sizeof(finit)); + out->write((const char*)&dgtest,sizeof(dgtest)); + out->write((const char*)&width,sizeof(width)); + out->write((const char*)&width1,sizeof(width1)); + out->write((const char*)&stx,sizeof(stx)); + out->write((const char*)&fx,sizeof(fx)); + out->write((const char*)&dgx,sizeof(dgx)); + out->write((const char*)&sty,sizeof(sty)); + out->write((const char*)&fy,sizeof(fy)); + out->write((const char*)&dgy,sizeof(dgy)); + out->write((const char*)&stmin,sizeof(stmin)); + out->write((const char*)&stmax,sizeof(stmax)); + } + + void deserialize(std::istream* in) const { + in->read((char*)&infoc, sizeof(infoc)); + in->read((char*)&dginit, sizeof(dginit)); + in->read((char*)&brackt, sizeof(brackt)); + in->read((char*)&stage1, sizeof(stage1)); + in->read((char*)&finit, sizeof(finit)); + in->read((char*)&dgtest, sizeof(dgtest)); + in->read((char*)&width, sizeof(width)); + in->read((char*)&width1, sizeof(width1)); + in->read((char*)&stx, sizeof(stx)); + in->read((char*)&fx, sizeof(fx)); + in->read((char*)&dgx, sizeof(dgx)); + in->read((char*)&sty, sizeof(sty)); + in->read((char*)&fy, sizeof(fy)); + in->read((char*)&dgy, sizeof(dgy)); + in->read((char*)&stmin, sizeof(stmin)); + in->read((char*)&stmax, sizeof(stmax)); + } + }; + + template <typename FloatType, typename SizeType> + void mcsrch<FloatType, SizeType>::run( + FloatType const& gtol, + FloatType const& stpmin, + FloatType const& stpmax, + SizeType n, + FloatType* x, + FloatType f, + const FloatType* g, + FloatType* s, + SizeType is0, + FloatType& stp, + FloatType ftol, + FloatType xtol, + SizeType maxfev, + int& info, + SizeType& nfev, + FloatType* wa) + { + if (info != -1) { + infoc = 1; + if ( n == 0 + || maxfev == 0 + || gtol < FloatType(0) + || xtol < FloatType(0) + || stpmin < FloatType(0) + || stpmax < stpmin) { + throw error_internal_error(__FILE__, __LINE__); + } + if (stp <= FloatType(0) || ftol < FloatType(0)) { + throw error_internal_error(__FILE__, __LINE__); + } + // Compute the initial gradient in the search direction + // and check that s is a descent direction. + dginit = FloatType(0); + for (SizeType j = 0; j < n; j++) { + dginit += g[j] * s[is0+j]; + } + if (dginit >= FloatType(0)) { + throw error_search_direction_not_descent(); + } + brackt = false; + stage1 = true; + nfev = 0; + finit = f; + dgtest = ftol*dginit; + width = stpmax - stpmin; + width1 = FloatType(2) * width; + std::copy(x, x+n, wa); + // The variables stx, fx, dgx contain the values of the step, + // function, and directional derivative at the best step. + // The variables sty, fy, dgy contain the value of the step, + // function, and derivative at the other endpoint of + // the interval of uncertainty. + // The variables stp, f, dg contain the values of the step, + // function, and derivative at the current step. + stx = FloatType(0); + fx = finit; + dgx = dginit; + sty = FloatType(0); + fy = finit; + dgy = dginit; + } + for (;;) { + if (info != -1) { + // Set the minimum and maximum steps to correspond + // to the present interval of uncertainty. + if (brackt) { + stmin = std::min(stx, sty); + stmax = std::max(stx, sty); + } + else { + stmin = stx; + stmax = stp + FloatType(4) * (stp - stx); + } + // Force the step to be within the bounds stpmax and stpmin. + stp = std::max(stp, stpmin); + stp = std::min(stp, stpmax); + // If an unusual termination is to occur then let + // stp be the lowest point obtained so far. + if ( (brackt && (stp <= stmin || stp >= stmax)) + || nfev >= maxfev - 1 || infoc == 0 + || (brackt && stmax - stmin <= xtol * stmax)) { + stp = stx; + } + // Evaluate the function and gradient at stp + // and compute the directional derivative. + // We return to main program to obtain F and G. + for (SizeType j = 0; j < n; j++) { + x[j] = wa[j] + stp * s[is0+j]; + } + info=-1; + break; + } + info = 0; + nfev++; + FloatType dg(0); + for (SizeType j = 0; j < n; j++) { + dg += g[j] * s[is0+j]; + } + FloatType ftest1 = finit + stp*dgtest; + // Test for convergence. + if ((brackt && (stp <= stmin || stp >= stmax)) || infoc == 0) { + throw error_line_search_failed_rounding_errors( + "Rounding errors prevent further progress." + " There may not be a step which satisfies the" + " sufficient decrease and curvature conditions." + " Tolerances may be too small."); + } + if (stp == stpmax && f <= ftest1 && dg <= dgtest) { + throw error_line_search_failed( + "The step is at the upper bound stpmax()."); + } + if (stp == stpmin && (f > ftest1 || dg >= dgtest)) { + throw error_line_search_failed( + "The step is at the lower bound stpmin()."); + } + if (nfev >= maxfev) { + throw error_line_search_failed( + "Number of function evaluations has reached maxfev()."); + } + if (brackt && stmax - stmin <= xtol * stmax) { + throw error_line_search_failed( + "Relative width of the interval of uncertainty" + " is at most xtol()."); + } + // Check for termination. + if (f <= ftest1 && abs(dg) <= gtol * (-dginit)) { + info = 1; + break; + } + // In the first stage we seek a step for which the modified + // function has a nonpositive value and nonnegative derivative. + if ( stage1 && f <= ftest1 + && dg >= std::min(ftol, gtol) * dginit) { + stage1 = false; + } + // A modified function is used to predict the step only if + // we have not obtained a step for which the modified + // function has a nonpositive function value and nonnegative + // derivative, and if a lower function value has been + // obtained but the decrease is not sufficient. + if (stage1 && f <= fx && f > ftest1) { + // Define the modified function and derivative values. + FloatType fm = f - stp*dgtest; + FloatType fxm = fx - stx*dgtest; + FloatType fym = fy - sty*dgtest; + FloatType dgm = dg - dgtest; + FloatType dgxm = dgx - dgtest; + FloatType dgym = dgy - dgtest; + // Call cstep to update the interval of uncertainty + // and to compute the new step. + infoc = mcstep(stx, fxm, dgxm, sty, fym, dgym, stp, fm, dgm, + brackt, stmin, stmax); + // Reset the function and gradient values for f. + fx = fxm + stx*dgtest; + fy = fym + sty*dgtest; + dgx = dgxm + dgtest; + dgy = dgym + dgtest; + } + else { + // Call mcstep to update the interval of uncertainty + // and to compute the new step. + infoc = mcstep(stx, fx, dgx, sty, fy, dgy, stp, f, dg, + brackt, stmin, stmax); + } + // Force a sufficient decrease in the size of the + // interval of uncertainty. + if (brackt) { + if (abs(sty - stx) >= FloatType(0.66) * width1) { + stp = stx + FloatType(0.5) * (sty - stx); + } + width1 = width; + width = abs(sty - stx); + } + } + } + + template <typename FloatType, typename SizeType> + int mcsrch<FloatType, SizeType>::mcstep( + FloatType& stx, + FloatType& fx, + FloatType& dx, + FloatType& sty, + FloatType& fy, + FloatType& dy, + FloatType& stp, + FloatType fp, + FloatType dp, + bool& brackt, + FloatType stpmin, + FloatType stpmax) + { + bool bound; + FloatType gamma, p, q, r, s, sgnd, stpc, stpf, stpq, theta; + int info = 0; + if ( ( brackt && (stp <= std::min(stx, sty) + || stp >= std::max(stx, sty))) + || dx * (stp - stx) >= FloatType(0) || stpmax < stpmin) { + return 0; + } + // Determine if the derivatives have opposite sign. + sgnd = dp * (dx / abs(dx)); + if (fp > fx) { + // First case. A higher function value. + // The minimum is bracketed. If the cubic step is closer + // to stx than the quadratic step, the cubic step is taken, + // else the average of the cubic and quadratic steps is taken. + info = 1; + bound = true; + theta = FloatType(3) * (fx - fp) / (stp - stx) + dx + dp; + s = max3(abs(theta), abs(dx), abs(dp)); + gamma = s * std::sqrt(pow2(theta / s) - (dx / s) * (dp / s)); + if (stp < stx) gamma = - gamma; + p = (gamma - dx) + theta; + q = ((gamma - dx) + gamma) + dp; + r = p/q; + stpc = stx + r * (stp - stx); + stpq = stx + + ((dx / ((fx - fp) / (stp - stx) + dx)) / FloatType(2)) + * (stp - stx); + if (abs(stpc - stx) < abs(stpq - stx)) { + stpf = stpc; + } + else { + stpf = stpc + (stpq - stpc) / FloatType(2); + } + brackt = true; + } + else if (sgnd < FloatType(0)) { + // Second case. A lower function value and derivatives of + // opposite sign. The minimum is bracketed. If the cubic + // step is closer to stx than the quadratic (secant) step, + // the cubic step is taken, else the quadratic step is taken. + info = 2; + bound = false; + theta = FloatType(3) * (fx - fp) / (stp - stx) + dx + dp; + s = max3(abs(theta), abs(dx), abs(dp)); + gamma = s * std::sqrt(pow2(theta / s) - (dx / s) * (dp / s)); + if (stp > stx) gamma = - gamma; + p = (gamma - dp) + theta; + q = ((gamma - dp) + gamma) + dx; + r = p/q; + stpc = stp + r * (stx - stp); + stpq = stp + (dp / (dp - dx)) * (stx - stp); + if (abs(stpc - stp) > abs(stpq - stp)) { + stpf = stpc; + } + else { + stpf = stpq; + } + brackt = true; + } + else if (abs(dp) < abs(dx)) { + // Third case. A lower function value, derivatives of the + // same sign, and the magnitude of the derivative decreases. + // The cubic step is only used if the cubic tends to infinity + // in the direction of the step or if the minimum of the cubic + // is beyond stp. Otherwise the cubic step is defined to be + // either stpmin or stpmax. The quadratic (secant) step is also + // computed and if the minimum is bracketed then the the step + // closest to stx is taken, else the step farthest away is taken. + info = 3; + bound = true; + theta = FloatType(3) * (fx - fp) / (stp - stx) + dx + dp; + s = max3(abs(theta), abs(dx), abs(dp)); + gamma = s * std::sqrt( + std::max(FloatType(0), pow2(theta / s) - (dx / s) * (dp / s))); + if (stp > stx) gamma = -gamma; + p = (gamma - dp) + theta; + q = (gamma + (dx - dp)) + gamma; + r = p/q; + if (r < FloatType(0) && gamma != FloatType(0)) { + stpc = stp + r * (stx - stp); + } + else if (stp > stx) { + stpc = stpmax; + } + else { + stpc = stpmin; + } + stpq = stp + (dp / (dp - dx)) * (stx - stp); + if (brackt) { + if (abs(stp - stpc) < abs(stp - stpq)) { + stpf = stpc; + } + else { + stpf = stpq; + } + } + else { + if (abs(stp - stpc) > abs(stp - stpq)) { + stpf = stpc; + } + else { + stpf = stpq; + } + } + } + else { + // Fourth case. A lower function value, derivatives of the + // same sign, and the magnitude of the derivative does + // not decrease. If the minimum is not bracketed, the step + // is either stpmin or stpmax, else the cubic step is taken. + info = 4; + bound = false; + if (brackt) { + theta = FloatType(3) * (fp - fy) / (sty - stp) + dy + dp; + s = max3(abs(theta), abs(dy), abs(dp)); + gamma = s * std::sqrt(pow2(theta / s) - (dy / s) * (dp / s)); + if (stp > sty) gamma = -gamma; + p = (gamma - dp) + theta; + q = ((gamma - dp) + gamma) + dy; + r = p/q; + stpc = stp + r * (sty - stp); + stpf = stpc; + } + else if (stp > stx) { + stpf = stpmax; + } + else { + stpf = stpmin; + } + } + // Update the interval of uncertainty. This update does not + // depend on the new step or the case analysis above. + if (fp > fx) { + sty = stp; + fy = fp; + dy = dp; + } + else { + if (sgnd < FloatType(0)) { + sty = stx; + fy = fx; + dy = dx; + } + stx = stp; + fx = fp; + dx = dp; + } + // Compute the new step and safeguard it. + stpf = std::min(stpmax, stpf); + stpf = std::max(stpmin, stpf); + stp = stpf; + if (brackt && bound) { + if (sty > stx) { + stp = std::min(stx + FloatType(0.66) * (sty - stx), stp); + } + else { + stp = std::max(stx + FloatType(0.66) * (sty - stx), stp); + } + } + return info; + } + + /* Compute the sum of a vector times a scalar plus another vector. + Adapted from the subroutine <code>daxpy</code> in + <code>lbfgs.f</code>. + */ + template <typename FloatType, typename SizeType> + void daxpy( + SizeType n, + FloatType da, + const FloatType* dx, + SizeType ix0, + SizeType incx, + FloatType* dy, + SizeType iy0, + SizeType incy) + { + SizeType i, ix, iy, m; + if (n == 0) return; + if (da == FloatType(0)) return; + if (!(incx == 1 && incy == 1)) { + ix = 0; + iy = 0; + for (i = 0; i < n; i++) { + dy[iy0+iy] += da * dx[ix0+ix]; + ix += incx; + iy += incy; + } + return; + } + m = n % 4; + for (i = 0; i < m; i++) { + dy[iy0+i] += da * dx[ix0+i]; + } + for (; i < n;) { + dy[iy0+i] += da * dx[ix0+i]; i++; + dy[iy0+i] += da * dx[ix0+i]; i++; + dy[iy0+i] += da * dx[ix0+i]; i++; + dy[iy0+i] += da * dx[ix0+i]; i++; + } + } + + template <typename FloatType, typename SizeType> + inline + void daxpy( + SizeType n, + FloatType da, + const FloatType* dx, + SizeType ix0, + FloatType* dy) + { + daxpy(n, da, dx, ix0, SizeType(1), dy, SizeType(0), SizeType(1)); + } + + /* Compute the dot product of two vectors. + Adapted from the subroutine <code>ddot</code> + in <code>lbfgs.f</code>. + */ + template <typename FloatType, typename SizeType> + FloatType ddot( + SizeType n, + const FloatType* dx, + SizeType ix0, + SizeType incx, + const FloatType* dy, + SizeType iy0, + SizeType incy) + { + SizeType i, ix, iy, m; + FloatType dtemp(0); + if (n == 0) return FloatType(0); + if (!(incx == 1 && incy == 1)) { + ix = 0; + iy = 0; + for (i = 0; i < n; i++) { + dtemp += dx[ix0+ix] * dy[iy0+iy]; + ix += incx; + iy += incy; + } + return dtemp; + } + m = n % 5; + for (i = 0; i < m; i++) { + dtemp += dx[ix0+i] * dy[iy0+i]; + } + for (; i < n;) { + dtemp += dx[ix0+i] * dy[iy0+i]; i++; + dtemp += dx[ix0+i] * dy[iy0+i]; i++; + dtemp += dx[ix0+i] * dy[iy0+i]; i++; + dtemp += dx[ix0+i] * dy[iy0+i]; i++; + dtemp += dx[ix0+i] * dy[iy0+i]; i++; + } + return dtemp; + } + + template <typename FloatType, typename SizeType> + inline + FloatType ddot( + SizeType n, + const FloatType* dx, + const FloatType* dy) + { + return ddot( + n, dx, SizeType(0), SizeType(1), dy, SizeType(0), SizeType(1)); + } + + } // namespace detail + + //! Interface to the LBFGS %minimizer. + /*! This class solves the unconstrained minimization problem + <pre> + min f(x), x = (x1,x2,...,x_n), + </pre> + using the limited-memory BFGS method. The routine is + especially effective on problems involving a large number of + variables. In a typical iteration of this method an + approximation Hk to the inverse of the Hessian + is obtained by applying <code>m</code> BFGS updates to a + diagonal matrix Hk0, using information from the + previous <code>m</code> steps. The user specifies the number + <code>m</code>, which determines the amount of storage + required by the routine. The user may also provide the + diagonal matrices Hk0 (parameter <code>diag</code> in the run() + function) if not satisfied with the default choice. The + algorithm is described in "On the limited memory BFGS method for + large scale optimization", by D. Liu and J. Nocedal, Mathematical + Programming B 45 (1989) 503-528. + + The user is required to calculate the function value + <code>f</code> and its gradient <code>g</code>. In order to + allow the user complete control over these computations, + reverse communication is used. The routine must be called + repeatedly under the control of the member functions + <code>requests_f_and_g()</code>, + <code>requests_diag()</code>. + If neither requests_f_and_g() nor requests_diag() is + <code>true</code> the user should check for convergence + (using class traditional_convergence_test or any + other custom test). If the convergence test is negative, + the minimizer may be called again for the next iteration. + + The steplength (stp()) is determined at each iteration + by means of the line search routine <code>mcsrch</code>, which is + a slight modification of the routine <code>CSRCH</code> written + by More' and Thuente. + + The only variables that are machine-dependent are + <code>xtol</code>, + <code>stpmin</code> and + <code>stpmax</code>. + + Fatal errors cause <code>error</code> exceptions to be thrown. + The generic class <code>error</code> is sub-classed (e.g. + class <code>error_line_search_failed</code>) to facilitate + granular %error handling. + + A note on performance: Using Compaq Fortran V5.4 and + Compaq C++ V6.5, the C++ implementation is about 15% slower + than the Fortran implementation. + */ + template <typename FloatType, typename SizeType = std::size_t> + class minimizer + { + public: + //! Default constructor. Some members are not initialized! + minimizer() + : n_(0), m_(0), maxfev_(0), + gtol_(0), xtol_(0), + stpmin_(0), stpmax_(0), + ispt(0), iypt(0) + {} + + //! Constructor. + /*! @param n The number of variables in the minimization problem. + Restriction: <code>n > 0</code>. + + @param m The number of corrections used in the BFGS update. + Values of <code>m</code> less than 3 are not recommended; + large values of <code>m</code> will result in excessive + computing time. <code>3 <= m <= 7</code> is + recommended. + Restriction: <code>m > 0</code>. + + @param maxfev Maximum number of function evaluations + <b>per line search</b>. + Termination occurs when the number of evaluations + of the objective function is at least <code>maxfev</code> by + the end of an iteration. + + @param gtol Controls the accuracy of the line search. + If the function and gradient evaluations are inexpensive with + respect to the cost of the iteration (which is sometimes the + case when solving very large problems) it may be advantageous + to set <code>gtol</code> to a small value. A typical small + value is 0.1. + Restriction: <code>gtol</code> should be greater than 1e-4. + + @param xtol An estimate of the machine precision (e.g. 10e-16 + on a SUN station 3/60). The line search routine will + terminate if the relative width of the interval of + uncertainty is less than <code>xtol</code>. + + @param stpmin Specifies the lower bound for the step + in the line search. + The default value is 1e-20. This value need not be modified + unless the exponent is too large for the machine being used, + or unless the problem is extremely badly scaled (in which + case the exponent should be increased). + + @param stpmax specifies the upper bound for the step + in the line search. + The default value is 1e20. This value need not be modified + unless the exponent is too large for the machine being used, + or unless the problem is extremely badly scaled (in which + case the exponent should be increased). + */ + explicit + minimizer( + SizeType n, + SizeType m = 5, + SizeType maxfev = 20, + FloatType gtol = FloatType(0.9), + FloatType xtol = FloatType(1.e-16), + FloatType stpmin = FloatType(1.e-20), + FloatType stpmax = FloatType(1.e20)) + : n_(n), m_(m), maxfev_(maxfev), + gtol_(gtol), xtol_(xtol), + stpmin_(stpmin), stpmax_(stpmax), + iflag_(0), requests_f_and_g_(false), requests_diag_(false), + iter_(0), nfun_(0), stp_(0), + stp1(0), ftol(0.0001), ys(0), point(0), npt(0), + ispt(n+2*m), iypt((n+2*m)+n*m), + info(0), bound(0), nfev(0) + { + if (n_ == 0) { + throw error_improper_input_parameter("n = 0."); + } + if (m_ == 0) { + throw error_improper_input_parameter("m = 0."); + } + if (maxfev_ == 0) { + throw error_improper_input_parameter("maxfev = 0."); + } + if (gtol_ <= FloatType(1.e-4)) { + throw error_improper_input_parameter("gtol <= 1.e-4."); + } + if (xtol_ < FloatType(0)) { + throw error_improper_input_parameter("xtol < 0."); + } + if (stpmin_ < FloatType(0)) { + throw error_improper_input_parameter("stpmin < 0."); + } + if (stpmax_ < stpmin) { + throw error_improper_input_parameter("stpmax < stpmin"); + } + w_.resize(n_*(2*m_+1)+2*m_); + scratch_array_.resize(n_); + } + + //! Number of free parameters (as passed to the constructor). + SizeType n() const { return n_; } + + //! Number of corrections kept (as passed to the constructor). + SizeType m() const { return m_; } + + /*! \brief Maximum number of evaluations of the objective function + per line search (as passed to the constructor). + */ + SizeType maxfev() const { return maxfev_; } + + /*! \brief Control of the accuracy of the line search. + (as passed to the constructor). + */ + FloatType gtol() const { return gtol_; } + + //! Estimate of the machine precision (as passed to the constructor). + FloatType xtol() const { return xtol_; } + + /*! \brief Lower bound for the step in the line search. + (as passed to the constructor). + */ + FloatType stpmin() const { return stpmin_; } + + /*! \brief Upper bound for the step in the line search. + (as passed to the constructor). + */ + FloatType stpmax() const { return stpmax_; } + + //! Status indicator for reverse communication. + /*! <code>true</code> if the run() function returns to request + evaluation of the objective function (<code>f</code>) and + gradients (<code>g</code>) for the current point + (<code>x</code>). To continue the minimization the + run() function is called again with the updated values for + <code>f</code> and <code>g</code>. + <p> + See also: requests_diag() + */ + bool requests_f_and_g() const { return requests_f_and_g_; } + + //! Status indicator for reverse communication. + /*! <code>true</code> if the run() function returns to request + evaluation of the diagonal matrix (<code>diag</code>) + for the current point (<code>x</code>). + To continue the minimization the run() function is called + again with the updated values for <code>diag</code>. + <p> + See also: requests_f_and_g() + */ + bool requests_diag() const { return requests_diag_; } + + //! Number of iterations so far. + /*! Note that one iteration may involve multiple evaluations + of the objective function. + <p> + See also: nfun() + */ + SizeType iter() const { return iter_; } + + //! Total number of evaluations of the objective function so far. + /*! The total number of function evaluations increases by the + number of evaluations required for the line search. The total + is only increased after a successful line search. + <p> + See also: iter() + */ + SizeType nfun() const { return nfun_; } + + //! Norm of gradient given gradient array of length n(). + FloatType euclidean_norm(const FloatType* a) const { + return std::sqrt(detail::ddot(n_, a, a)); + } + + //! Current stepsize. + FloatType stp() const { return stp_; } + + //! Execution of one step of the minimization. + /*! @param x On initial entry this must be set by the user to + the values of the initial estimate of the solution vector. + + @param f Before initial entry or on re-entry under the + control of requests_f_and_g(), <code>f</code> must be set + by the user to contain the value of the objective function + at the current point <code>x</code>. + + @param g Before initial entry or on re-entry under the + control of requests_f_and_g(), <code>g</code> must be set + by the user to contain the components of the gradient at + the current point <code>x</code>. + + The return value is <code>true</code> if either + requests_f_and_g() or requests_diag() is <code>true</code>. + Otherwise the user should check for convergence + (e.g. using class traditional_convergence_test) and + call the run() function again to continue the minimization. + If the return value is <code>false</code> the user + should <b>not</b> update <code>f</code>, <code>g</code> or + <code>diag</code> (other overload) before calling + the run() function again. + + Note that <code>x</code> is always modified by the run() + function. Depending on the situation it can therefore be + necessary to evaluate the objective function one more time + after the minimization is terminated. + */ + bool run( + FloatType* x, + FloatType f, + const FloatType* g) + { + return generic_run(x, f, g, false, 0); + } + + //! Execution of one step of the minimization. + /*! @param x See other overload. + + @param f See other overload. + + @param g See other overload. + + @param diag On initial entry or on re-entry under the + control of requests_diag(), <code>diag</code> must be set by + the user to contain the values of the diagonal matrix Hk0. + The routine will return at each iteration of the algorithm + with requests_diag() set to <code>true</code>. + <p> + Restriction: all elements of <code>diag</code> must be + positive. + */ + bool run( + FloatType* x, + FloatType f, + const FloatType* g, + const FloatType* diag) + { + return generic_run(x, f, g, true, diag); + } + + void serialize(std::ostream* out) const { + out->write((const char*)&n_, sizeof(n_)); // sanity check + out->write((const char*)&m_, sizeof(m_)); // sanity check + SizeType fs = sizeof(FloatType); + out->write((const char*)&fs, sizeof(fs)); // sanity check + + mcsrch_instance.serialize(out); + out->write((const char*)&iflag_, sizeof(iflag_)); + out->write((const char*)&requests_f_and_g_, sizeof(requests_f_and_g_)); + out->write((const char*)&requests_diag_, sizeof(requests_diag_)); + out->write((const char*)&iter_, sizeof(iter_)); + out->write((const char*)&nfun_, sizeof(nfun_)); + out->write((const char*)&stp_, sizeof(stp_)); + out->write((const char*)&stp1, sizeof(stp1)); + out->write((const char*)&ftol, sizeof(ftol)); + out->write((const char*)&ys, sizeof(ys)); + out->write((const char*)&point, sizeof(point)); + out->write((const char*)&npt, sizeof(npt)); + out->write((const char*)&info, sizeof(info)); + out->write((const char*)&bound, sizeof(bound)); + out->write((const char*)&nfev, sizeof(nfev)); + out->write((const char*)&w_[0], sizeof(FloatType) * w_.size()); + out->write((const char*)&scratch_array_[0], sizeof(FloatType) * scratch_array_.size()); + } + + void deserialize(std::istream* in) { + SizeType n, m, fs; + in->read((char*)&n, sizeof(n)); + in->read((char*)&m, sizeof(m)); + in->read((char*)&fs, sizeof(fs)); + assert(n == n_); + assert(m == m_); + assert(fs == sizeof(FloatType)); + + mcsrch_instance.deserialize(in); + in->read((char*)&iflag_, sizeof(iflag_)); + in->read((char*)&requests_f_and_g_, sizeof(requests_f_and_g_)); + in->read((char*)&requests_diag_, sizeof(requests_diag_)); + in->read((char*)&iter_, sizeof(iter_)); + in->read((char*)&nfun_, sizeof(nfun_)); + in->read((char*)&stp_, sizeof(stp_)); + in->read((char*)&stp1, sizeof(stp1)); + in->read((char*)&ftol, sizeof(ftol)); + in->read((char*)&ys, sizeof(ys)); + in->read((char*)&point, sizeof(point)); + in->read((char*)&npt, sizeof(npt)); + in->read((char*)&info, sizeof(info)); + in->read((char*)&bound, sizeof(bound)); + in->read((char*)&nfev, sizeof(nfev)); + in->read((char*)&w_[0], sizeof(FloatType) * w_.size()); + in->read((char*)&scratch_array_[0], sizeof(FloatType) * scratch_array_.size()); + } + + protected: + static void throw_diagonal_element_not_positive(SizeType i) { + throw error_improper_input_data( + "The " + error::itoa(i) + ". diagonal element of the" + " inverse Hessian approximation is not positive."); + } + + bool generic_run( + FloatType* x, + FloatType f, + const FloatType* g, + bool diagco, + const FloatType* diag); + + detail::mcsrch<FloatType, SizeType> mcsrch_instance; + const SizeType n_; + const SizeType m_; + const SizeType maxfev_; + const FloatType gtol_; + const FloatType xtol_; + const FloatType stpmin_; + const FloatType stpmax_; + int iflag_; + bool requests_f_and_g_; + bool requests_diag_; + SizeType iter_; + SizeType nfun_; + FloatType stp_; + FloatType stp1; + FloatType ftol; + FloatType ys; + SizeType point; + SizeType npt; + const SizeType ispt; + const SizeType iypt; + int info; + SizeType bound; + SizeType nfev; + std::vector<FloatType> w_; + std::vector<FloatType> scratch_array_; + }; + + template <typename FloatType, typename SizeType> + bool minimizer<FloatType, SizeType>::generic_run( + FloatType* x, + FloatType f, + const FloatType* g, + bool diagco, + const FloatType* diag) + { + bool execute_entire_while_loop = false; + if (!(requests_f_and_g_ || requests_diag_)) { + execute_entire_while_loop = true; + } + requests_f_and_g_ = false; + requests_diag_ = false; + FloatType* w = &(*(w_.begin())); + if (iflag_ == 0) { // Initialize. + nfun_ = 1; + if (diagco) { + for (SizeType i = 0; i < n_; i++) { + if (diag[i] <= FloatType(0)) { + throw_diagonal_element_not_positive(i); + } + } + } + else { + std::fill_n(scratch_array_.begin(), n_, FloatType(1)); + diag = &(*(scratch_array_.begin())); + } + for (SizeType i = 0; i < n_; i++) { + w[ispt + i] = -g[i] * diag[i]; + } + FloatType gnorm = std::sqrt(detail::ddot(n_, g, g)); + if (gnorm == FloatType(0)) return false; + stp1 = FloatType(1) / gnorm; + execute_entire_while_loop = true; + } + if (execute_entire_while_loop) { + bound = iter_; + iter_++; + info = 0; + if (iter_ != 1) { + if (iter_ > m_) bound = m_; + ys = detail::ddot( + n_, w, iypt + npt, SizeType(1), w, ispt + npt, SizeType(1)); + if (!diagco) { + FloatType yy = detail::ddot( + n_, w, iypt + npt, SizeType(1), w, iypt + npt, SizeType(1)); + std::fill_n(scratch_array_.begin(), n_, ys / yy); + diag = &(*(scratch_array_.begin())); + } + else { + iflag_ = 2; + requests_diag_ = true; + return true; + } + } + } + if (execute_entire_while_loop || iflag_ == 2) { + if (iter_ != 1) { + if (diag == 0) { + throw error_internal_error(__FILE__, __LINE__); + } + if (diagco) { + for (SizeType i = 0; i < n_; i++) { + if (diag[i] <= FloatType(0)) { + throw_diagonal_element_not_positive(i); + } + } + } + SizeType cp = point; + if (point == 0) cp = m_; + w[n_ + cp -1] = 1 / ys; + SizeType i; + for (i = 0; i < n_; i++) { + w[i] = -g[i]; + } + cp = point; + for (i = 0; i < bound; i++) { + if (cp == 0) cp = m_; + cp--; + FloatType sq = detail::ddot( + n_, w, ispt + cp * n_, SizeType(1), w, SizeType(0), SizeType(1)); + SizeType inmc=n_+m_+cp; + SizeType iycn=iypt+cp*n_; + w[inmc] = w[n_ + cp] * sq; + detail::daxpy(n_, -w[inmc], w, iycn, w); + } + for (i = 0; i < n_; i++) { + w[i] *= diag[i]; + } + for (i = 0; i < bound; i++) { + FloatType yr = detail::ddot( + n_, w, iypt + cp * n_, SizeType(1), w, SizeType(0), SizeType(1)); + FloatType beta = w[n_ + cp] * yr; + SizeType inmc=n_+m_+cp; + beta = w[inmc] - beta; + SizeType iscn=ispt+cp*n_; + detail::daxpy(n_, beta, w, iscn, w); + cp++; + if (cp == m_) cp = 0; + } + std::copy(w, w+n_, w+(ispt + point * n_)); + } + stp_ = FloatType(1); + if (iter_ == 1) stp_ = stp1; + std::copy(g, g+n_, w); + } + mcsrch_instance.run( + gtol_, stpmin_, stpmax_, n_, x, f, g, w, ispt + point * n_, + stp_, ftol, xtol_, maxfev_, info, nfev, &(*(scratch_array_.begin()))); + if (info == -1) { + iflag_ = 1; + requests_f_and_g_ = true; + return true; + } + if (info != 1) { + throw error_internal_error(__FILE__, __LINE__); + } + nfun_ += nfev; + npt = point*n_; + for (SizeType i = 0; i < n_; i++) { + w[ispt + npt + i] = stp_ * w[ispt + npt + i]; + w[iypt + npt + i] = g[i] - w[i]; + } + point++; + if (point == m_) point = 0; + return false; + } + + //! Traditional LBFGS convergence test. + /*! This convergence test is equivalent to the test embedded + in the <code>lbfgs.f</code> Fortran code. The test assumes that + there is a meaningful relation between the Euclidean norm of the + parameter vector <code>x</code> and the norm of the gradient + vector <code>g</code>. Therefore this test should not be used if + this assumption is not correct for a given problem. + */ + template <typename FloatType, typename SizeType = std::size_t> + class traditional_convergence_test + { + public: + //! Default constructor. + traditional_convergence_test() + : n_(0), eps_(0) + {} + + //! Constructor. + /*! @param n The number of variables in the minimization problem. + Restriction: <code>n > 0</code>. + + @param eps Determines the accuracy with which the solution + is to be found. + */ + explicit + traditional_convergence_test( + SizeType n, + FloatType eps = FloatType(1.e-5)) + : n_(n), eps_(eps) + { + if (n_ == 0) { + throw error_improper_input_parameter("n = 0."); + } + if (eps_ < FloatType(0)) { + throw error_improper_input_parameter("eps < 0."); + } + } + + //! Number of free parameters (as passed to the constructor). + SizeType n() const { return n_; } + + /*! \brief Accuracy with which the solution is to be found + (as passed to the constructor). + */ + FloatType eps() const { return eps_; } + + //! Execution of the convergence test for the given parameters. + /*! Returns <code>true</code> if + <pre> + ||g|| < eps * max(1,||x||), + </pre> + where <code>||.||</code> denotes the Euclidean norm. + + @param x Current solution vector. + + @param g Components of the gradient at the current + point <code>x</code>. + */ + bool + operator()(const FloatType* x, const FloatType* g) const + { + FloatType xnorm = std::sqrt(detail::ddot(n_, x, x)); + FloatType gnorm = std::sqrt(detail::ddot(n_, g, g)); + if (gnorm <= eps_ * std::max(FloatType(1), xnorm)) return true; + return false; + } + protected: + const SizeType n_; + const FloatType eps_; + }; + +}} // namespace scitbx::lbfgs + +template <typename T> +std::ostream& operator<<(std::ostream& os, const scitbx::lbfgs::minimizer<T>& min) { + return os << "ITER=" << min.iter() << "\tNFUN=" << min.nfun() << "\tSTP=" << min.stp() << "\tDIAG=" << min.requests_diag() << "\tF&G=" << min.requests_f_and_g(); +} + + +#endif // SCITBX_LBFGS_H |