1690 lines
54 KiB
C++
1690 lines
54 KiB
C++
/*
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LA: linear algebra C++ interface library
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Copyright (C) 2026 Jiri Pittner <jiri.pittner@jh-inst.cas.cz> or <jiri@pittnerovi.com>
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This program is free software: you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation, either version 3 of the License, or
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(at your option) any later version.
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This program is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with this program. If not, see <http://www.gnu.org/licenses/>.
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*/
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#include "optimize.h"
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#include "la_random.h"
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#include "diis.h"
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#include "nonclass.h"
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namespace LA {
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template<typename T>
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T testproblem<T>::funcgrad(const NRVec<T> &x, NRVec<T> *grad) const
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{
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int n=x.size();
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T r= n*(n-1)/2;
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for(int i=0; i<n; ++i)
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{
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T x2=x[i]*x[i];
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r += (i+1)*(x2 + .1*x2*x2);
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for(int j=0; j<i; ++j) r -= cos(x[i]*x[j]);
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}
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if(grad)
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{
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if(grad->size()!=x.size()) laerror("grad size mismatch");
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grad->copyonwrite(); //I forgot this initially
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for(int k=0; k<n; ++k)
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{
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(*grad)[k] = (k+1)*x[k]*(2+0.4*x[k]*x[k]);
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for(int i=0; i<n; ++i) if(i!=k) (*grad)[k] += x[i]*sin(x[i]*x[k]);
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}
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}
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return r;
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}
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template<typename T>
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void testproblem_vqe_constrained<T>::constraints(const NRVec<T> &x, NRVec<T> &c) const
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{
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c.copyonwrite();
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c[0]=0;
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for(int i=0; i<x.size()/2; ++i) c[0]+= std::cos(x[2*i])*std::cos(x[2*i+1]);
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}
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//central difference (f(x+h e_i) - f(x-h e_i))/(2h); xs is a working copy of x, restored on return
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template<typename T>
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static T central_difference(const OptProblem<T> *p, NRVec<T> &xs, const NRVec<T> &x, const int i, const T h)
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{
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xs[i]=x[i]+h;
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T d=p->funcgrad(xs,NULL);
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xs[i]=x[i]-h;
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d-=p->funcgrad(xs,NULL);
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xs[i]=x[i];
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return d/(2*h);
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}
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template<typename T>
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ParameterShift<T>::ParameterShift(const OptProblem<T> *pp, const T hh) : OptProblemAdapter<T>(pp), trig(optcapability<OptTrigonometric<T> >(pp)), h(hh)
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{
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if(!trig) laerror("ParameterShift needs a problem providing OptTrigonometric");
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}
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//derivative of a trigonometric polynomial of degree R with frequencies 1..R (in units of w):
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//f'(0) = sum_{mu=1}^{2R} (-1)^{mu-1} f(s_mu)/(4R sin^2(s_mu/2)), s_mu = (2mu-1)pi/(2R) (Wierichs et al., Quantum 6, 677 (2022))
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template<typename T>
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T ParameterShift<T>::funcgrad(const NRVec<T> &x, NRVec<T> *grad) const
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{
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T f=this->p->funcgrad(x,NULL);
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if(!grad) return f;
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grad->copyonwrite();
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NRVec<T> xs(x);
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xs.copyonwrite(); //working copy with shifted elements
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for(int i=0; i<x.size(); ++i)
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{
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const int R=trig->degree(i);
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if(R==0) {(*grad)[i]=central_difference(this->p,xs,x,i,h); continue;}
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const T w=2*M_PI/trig->period(i);
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T d=0;
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for(int mu=1; mu<=2*R; ++mu)
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{
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const T s=(2*mu-1)*M_PI/(2*R);
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const T sn=std::sin(s/2);
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xs[i]=x[i]+s/w;
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d+= (mu%2 ? 1 : -1)*this->p->funcgrad(xs,NULL)/(4*R*sn*sn);
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}
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xs[i]=x[i];
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(*grad)[i]=w*d;
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}
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return f;
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}
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template<typename T>
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T FiniteDifference<T>::funcgrad(const NRVec<T> &x, NRVec<T> *grad) const
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{
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T f=this->p->funcgrad(x,NULL);
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if(!grad) return f;
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grad->copyonwrite();
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NRVec<T> xs(x);
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xs.copyonwrite(); //working copy with shifted elements
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for(int i=0; i<x.size(); ++i) (*grad)[i]=central_difference(this->p,xs,x,i,h);
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return f;
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}
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//Bloch vector components z = cos a cos b, x = cos a sin b of the qubit RY(b) RX(a)|0> and their derivatives
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template<typename T>
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class blochqubit {
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public:
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T z, x, za, zb, xa, xb;
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blochqubit(const T a, const T b)
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{
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T ca=std::cos(a), sa=std::sin(a), cb=std::cos(b), sb=std::sin(b);
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z=ca*cb; x=ca*sb; za= -sa*cb; zb= -ca*sb; xa= -sa*sb; xb=ca*cb;
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}
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};
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template<typename T>
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T testproblem_vqe<T>::funcgrad(const NRVec<T> &x, NRVec<T> *grad) const
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{
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const int nq=x.size()/2;
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if(2*nq!=x.size()) laerror("testproblem_vqe needs an even number of parameters");
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T r=0;
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for(int i=0; i<nq; ++i)
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{
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blochqubit<T> q(x[2*i],x[2*i+1]);
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r+= hfield(i)*q.z + kfield(i)*q.x;
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if(i<nq-1) r+= J(i)*q.z*std::cos(x[2*i+2])*std::cos(x[2*i+3]);
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}
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if(grad)
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{
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if(grad->size()!=x.size()) laerror("grad size mismatch");
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grad->copyonwrite();
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for(int i=0; i<nq; ++i)
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{
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blochqubit<T> q(x[2*i],x[2*i+1]);
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T eff=hfield(i); //coefficient of z_i including the couplings
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if(i>0) eff+= J(i-1)*std::cos(x[2*i-2])*std::cos(x[2*i-1]);
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if(i<nq-1) eff+= J(i)*std::cos(x[2*i+2])*std::cos(x[2*i+3]);
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(*grad)[2*i]= eff*q.za + kfield(i)*q.xa;
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(*grad)[2*i+1]= eff*q.zb + kfield(i)*q.xb;
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}
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}
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return r;
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}
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template<typename T>
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void testproblem_vqe<T>::hessian(const NRVec<T> &x, NRSMat<T> &h) const
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{
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const int nq=x.size()/2;
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h=(T)0;
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for(int i=0; i<nq; ++i)
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{
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blochqubit<T> q(x[2*i],x[2*i+1]);
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T eff=hfield(i);
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if(i>0) eff+= J(i-1)*std::cos(x[2*i-2])*std::cos(x[2*i-1]);
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if(i<nq-1) eff+= J(i)*std::cos(x[2*i+2])*std::cos(x[2*i+3]);
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//z_aa = z_bb = -z, x_aa = x_bb = -x, z_ab = sin a sin b, x_ab = -sin a cos b
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h(2*i,2*i)= -eff*q.z - kfield(i)*q.x;
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h(2*i+1,2*i+1)= h(2*i,2*i);
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h(2*i,2*i+1)= eff*std::sin(x[2*i])*std::sin(x[2*i+1]) - kfield(i)*std::sin(x[2*i])*std::cos(x[2*i+1]);
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if(i<nq-1)
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{
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blochqubit<T> q1(x[2*i+2],x[2*i+3]);
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h(2*i,2*i+2)= J(i)*q.za*q1.za;
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h(2*i,2*i+3)= J(i)*q.za*q1.zb;
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h(2*i+1,2*i+2)= J(i)*q.zb*q1.za;
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h(2*i+1,2*i+3)= J(i)*q.zb*q1.zb;
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}
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}
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}
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template<typename T>
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void testproblem_vqe<T>::metric(const NRVec<T> &x, NRSMat<T> &g) const
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{
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const int nq=x.size()/2;
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g=(T)0;
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for(int i=0; i<nq; ++i)
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{
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T ca=std::cos(x[2*i]);
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g(2*i,2*i)=0.25;
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g(2*i+1,2*i+1)=0.25*ca*ca;
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}
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}
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//safeguarded cubic interpolation of the minimizer of phi between a and b, given function values and derivatives at both ends
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template<typename T>
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static T cubic_interpolate(const T a, const T fa, const T da, const T b, const T fb, const T db)
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{
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T lo= a<b ? a : b;
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T hi= a<b ? b : a;
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T safe=0.1*(hi-lo); //keep the trial point away from the interval ends
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T d1 = da + db - 3*(fa-fb)/(a-b);
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T disc = d1*d1 - da*db;
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T r;
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if(disc<0) r=(a+b)/2;
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else
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{
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T d2 = std::sqrt(disc); if(b<a) d2 = -d2;
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T den = db - da + 2*d2;
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if(den==0) r=(a+b)/2;
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else r = b - (b-a)*(db + d2 - d1)/den;
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}
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if(!(r>=lo+safe && r<=hi-safe)) r=(a+b)/2; //also catches NaN
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return r;
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}
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template<typename T>
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T linesearch_wolfe(optevaluator<T> &e, NRVec<T> &x, T &f, NRVec<T> &g, const NRVec<T> &p, const T alpha1, const linesearch_hyperparameters<T> &lp)
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{
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const T c1=lp.my_c1;
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const T c2=lp.my_c2;
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const int verbose=lp.my_verbose;
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const int maxiter= std::min(lp.my_maxlinesearch,e.remaining());
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const int n=x.size();
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const NRVec<T> x0(x);
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const T f0=f;
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const T d0= g*p;
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if(d0>=0) {if(verbose) std::cout <<"linesearch: not a descent direction\n"; return 0;}
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const T alphamax=lp.my_maxsteplength/p.norm();
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T alpha=std::min(alpha1,alphamax);
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//data of trial points; vectors are always freshly allocated, so that assignments (which share data) are safe
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T alpha0=0, f_alpha0=f0, d_alpha0=d0; NRVec<T> x_alpha0(x0), g_alpha0(g); //previous trial point
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T alo=0,flo=0,dlo=0,ahi=0,fhi=0,dhi=0; NRVec<T> xlo,glo; //bracketing interval for the zoom phase, lo is the best point
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bool bracketed=false;
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int it;
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//bracketing phase
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for(it=0; it<maxiter; ++it)
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{
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NRVec<T> xt = x0 + p*alpha;
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NRVec<T> gt(n);
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T ft=e(xt,>);
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T dt= gt*p;
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if(verbose>1) std::cout <<"linesearch bracket "<<it<<" alpha= "<<alpha<<" f= "<<ft<<" dphi= "<<dt<<std::endl;
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if(ft > f0 + c1*alpha*d0 || (it>0 && ft>=f_alpha0))
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{
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alo=alpha0; flo=f_alpha0; dlo=d_alpha0; xlo=x_alpha0; glo=g_alpha0;
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ahi=alpha; fhi=ft; dhi=dt;
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bracketed=true;
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break;
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}
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if(std::abs(dt) <= -c2*d0 || alpha>=alphamax) {x=xt; f=ft; g=gt; return alpha;} //strong Wolfe satisfied or cannot go further
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if(dt>=0)
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{
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alo=alpha; flo=ft; dlo=dt; xlo=xt; glo=gt;
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ahi=alpha0; fhi=f_alpha0; dhi=d_alpha0;
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bracketed=true;
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break;
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}
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alpha0=alpha; f_alpha0=ft; d_alpha0=dt; x_alpha0=xt; g_alpha0=gt;
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alpha *= 2; if(alpha>alphamax) alpha=alphamax;
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}
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//zoom phase, invariant: alo satisfies sufficient decrease and has the lowest f found so far, dlo*(ahi-alo)<0
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if(bracketed) for(++it; it<maxiter; ++it)
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{
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T a=cubic_interpolate(alo,flo,dlo,ahi,fhi,dhi);
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NRVec<T> xt = x0 + p*a;
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NRVec<T> gt(n);
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T ft=e(xt,>);
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T dt= gt*p;
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if(verbose>1) std::cout <<"linesearch zoom "<<it<<" alpha= "<<a<<" f= "<<ft<<" dphi= "<<dt<<std::endl;
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if(ft > f0 + c1*a*d0 || ft>=flo)
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{
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ahi=a; fhi=ft; dhi=dt;
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}
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else
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{
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if(std::abs(dt) <= -c2*d0) {x=xt; f=ft; g=gt; return a;}
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if(dt*(ahi-alo)>=0) {ahi=alo; fhi=flo; dhi=dlo;}
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alo=a; flo=ft; dlo=dt; xlo=xt; glo=gt;
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}
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if(std::abs(ahi-alo) <= std::numeric_limits<T>::epsilon()*std::abs(alo)) break; //interval collapsed
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}
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//failure - accept the best point satisfying sufficient decrease, if any
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if(!bracketed && alpha0>0) {alo=alpha0; flo=f_alpha0; xlo=x_alpha0; glo=g_alpha0;} //bracketing ran out of evaluations
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if(alo>0)
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{
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if(verbose) std::cout <<"linesearch: strong Wolfe conditions not satisfied, accepting sufficient decrease step "<<alo<<std::endl;
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x=xlo; f=flo; g=glo; return alo;
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}
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if(verbose) std::cout <<"linesearch failed\n";
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return 0;
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}
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//quasi-Newton update of the inverse Hessian approximation H, returns false if the update was skipped
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//Hinit = H is still the initial guess, it is then scaled by s.y/y.y (Nocedal & Wright eq. 6.20)
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//for SR1 the scaling makes (s-Hy).y=0, so its first update is skipped and the next one starts from the scaled H
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//rank-1 terms are done via rank2update(a,x,x) = 2a x x^T
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template<typename T>
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static bool QN_update_H(NRSMat<T> &H, bool &Hinit, const NRVec<T> &s, const NRVec<T> &y, const BFGS_hyperparameters<T> &hp)
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{
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const T sy = s*y;
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const T snorm=s.norm();
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const T ynorm=y.norm();
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const bool curvature = sy > hp.my_sy_thr*snorm*ynorm;
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if(Hinit && curvature) {H = (T)(sy/(ynorm*ynorm)); Hinit=false;}
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switch(hp.my_update)
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{
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case QN_BFGS: //H+ = (I - rho s y^T) H (I - rho y s^T) + rho s s^T = H + s u^T + u s^T, rho=1/(y^T s), u = (rho^2 y^T H y + rho)/2 s - rho Hy
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{
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if(!curvature) return false;
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T rho = 1/sy;
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NRVec<T> Hy = H*y;
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NRVec<T> u = s*((rho*rho*(y*Hy) + rho)/2);
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u.axpy(-rho,Hy);
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H.rank2update((T)1,s,u);
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}
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break;
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case QN_DFP: //H+ = H - Hy (Hy)^T/(y^T H y) + s s^T/(y^T s)
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{
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if(!curvature) return false;
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NRVec<T> Hy = H*y;
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T yHy = y*Hy;
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if(yHy<=0) return false;
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H.rank2update(1/(2*sy),s,s);
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H.rank2update(-1/(2*yHy),Hy,Hy);
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}
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break;
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case QN_SR1: //H+ = H + w w^T/(w^T y), w = s - Hy; H may become indefinite
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{
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NRVec<T> w = H*y;
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w *= (T)-1;
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w += s;
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T wy = w*y;
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if(std::abs(wy) <= hp.my_sr1_thr*w.norm()*ynorm) return false;
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H.rank2update(1/(2*wy),w,w);
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}
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break;
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default: laerror("unknown quasi-Newton update");
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}
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Hinit=false;
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return true;
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}
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template<typename T>
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bool optstate<T>::stop(const T ff, const T gn, const int needed)
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{
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f=ff; gnorm=gn;
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if(hp.my_verbose)
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{
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std::cout <<name<<" iter "<<iter<<" f= "<<f;
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if(gnorm>=0) std::cout <<" |g|= "<<gnorm;
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std::cout <<std::endl;
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}
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if(hp.converged(*this)) {converged=true; return true;}
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if(iter>=hp.my_maxstep) {if(hp.my_verbose) std::cout <<name<<": maximum number of iterations reached\n"; return true;}
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if(e.remaining()<needed) {if(hp.my_verbose) std::cout <<name<<": maximum number of function evaluations reached\n"; return true;}
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return false;
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}
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template<typename T>
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optresult<T> optstate<T>::finish(hyperparameters<T> *achieved) const
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{
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if(hp.my_verbose)
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{
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std::cout <<name<<(converged?" converged in ":" NOT converged after ")<<iter<<" iterations, "<<e.nfunc<<" function evaluations, f= "<<f;
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if(gnorm>=0) std::cout <<" |g|= "<<gnorm;
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std::cout <<std::endl;
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}
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if(achieved) {achieved->setfrom(hp); achieved->setreached(*this);}
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return optresult<T>(f,converged);
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}
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//function along the line x0 + t*u for linemin_brent, keeps the best point and counts the evaluations
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template<typename T>
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class linefunction {
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public:
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optevaluator<T> &e;
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const NRVec<T> &x0;
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const NRVec<T> &u;
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int maxiter; //evaluation budget
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int nev; //evaluations done
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T tbest;
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T fbest;
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linefunction(optevaluator<T> &ee, const NRVec<T> &xx0, const T f0, const NRVec<T> &uu, const int maxit) : e(ee), x0(xx0), u(uu), maxiter(std::min(maxit,ee.remaining())), nev(0), tbest(0), fbest(f0) {};
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NRVec<T> point(const T t) const {NRVec<T> x(x0); x.axpy(t,u); return x;}
|
|
T operator()(const T t) {T f=e(point(t)); ++nev; if(f<fbest) {fbest=f; tbest=t;} return f;}
|
|
bool exhausted() const {return nev>=maxiter;}
|
|
};
|
|
|
|
|
|
template<typename T>
|
|
T linemin_brent(optevaluator<T> &e, NRVec<T> &x, T &f, const NRVec<T> &u, const T t1, const T tol, const T abstol, const int maxiter, const int verbose)
|
|
{
|
|
const T golden=1.618033988749895; //expansion factor of the bracketing
|
|
const T cgold=0.3819660112501051; //golden section fraction (3-sqrt(5))/2
|
|
linefunction<T> phi(e,x,f,u,maxiter);
|
|
|
|
//bracketing: a,b,c with f(b)<=f(a), f(b)<=f(c)
|
|
T a=0, fa=f;
|
|
T b=t1, fb;
|
|
T c=0, fc=0;
|
|
bool bracketed=false;
|
|
if(phi.exhausted()) return 0;
|
|
fb=phi(b);
|
|
if(fb>fa) //uphill, try the opposite direction
|
|
{
|
|
if(phi.exhausted()) return 0;
|
|
c= -t1; fc=phi(c);
|
|
if(fc>=fa) {std::swap(a,b); std::swap(fa,fb); bracketed=true;} //minimum between -t1 and t1, middle point is 0
|
|
else {b=c; fb=fc;}
|
|
}
|
|
if(!bracketed) //expand downhill from a through b
|
|
{
|
|
for(;;)
|
|
{
|
|
if(phi.exhausted()) break;
|
|
c= b+golden*(b-a); fc=phi(c);
|
|
if(fc>=fb) {bracketed=true; break;}
|
|
a=b; fa=fb; b=c; fb=fc;
|
|
}
|
|
}
|
|
if(verbose>1) std::cout <<"linemin bracket "<<a<<" "<<b<<" "<<c<<" f= "<<fa<<" "<<fb<<" "<<fc<<std::endl;
|
|
|
|
//Brent's method in [lo,hi]: xm = best point, w = second best, v = previous value of w
|
|
if(bracketed)
|
|
{
|
|
T lo=std::min(a,c), hi=std::max(a,c);
|
|
T xm=b, w=b, v=b, fx=fb, fw=fb, fv=fb;
|
|
T d=0, dprev=0; //last step and the one before
|
|
const T tolabs=std::max(abstol,std::numeric_limits<T>::epsilon()*std::abs(t1)); //nonzero to guarantee termination
|
|
while(!phi.exhausted())
|
|
{
|
|
T mid=(lo+hi)/2;
|
|
T tol1=tol*std::abs(xm)+tolabs;
|
|
T tol2=2*tol1;
|
|
if(std::abs(xm-mid) <= tol2-(hi-lo)/2) break; //converged
|
|
bool goldenstep=true;
|
|
if(std::abs(dprev)>tol1) //try parabolic interpolation through xm, w, v
|
|
{
|
|
T r=(xm-w)*(fx-fv);
|
|
T q=(xm-v)*(fx-fw);
|
|
T p=(xm-v)*q-(xm-w)*r;
|
|
q=2*(q-r);
|
|
if(q>0) p= -p; else q= -q;
|
|
if(std::abs(p)<std::abs(q*dprev/2) && p>q*(lo-xm) && p<q*(hi-xm)) //acceptable: inside the interval and less than half of the step before last
|
|
{
|
|
dprev=d;
|
|
d=p/q;
|
|
T t=xm+d;
|
|
if(t-lo<tol2 || hi-t<tol2) d= mid>xm ? tol1 : -tol1;
|
|
goldenstep=false;
|
|
}
|
|
}
|
|
if(goldenstep) {dprev= xm>=mid ? lo-xm : hi-xm; d=cgold*dprev;}
|
|
T t= std::abs(d)>=tol1 ? xm+d : xm+(d>0 ? tol1 : -tol1);
|
|
T ft=phi(t);
|
|
if(ft<=fx)
|
|
{
|
|
if(t>=xm) lo=xm; else hi=xm;
|
|
v=w; fv=fw; w=xm; fw=fx; xm=t; fx=ft;
|
|
}
|
|
else
|
|
{
|
|
if(t<xm) lo=t; else hi=t;
|
|
if(ft<=fw || w==xm) {v=w; fv=fw; w=t; fw=ft;}
|
|
else if(ft<=fv || v==xm || v==w) {v=t; fv=ft;}
|
|
}
|
|
}
|
|
}
|
|
|
|
if(verbose>1) std::cout <<"linemin t= "<<phi.tbest<<" f= "<<phi.fbest<<" evaluations "<<phi.nev<<std::endl;
|
|
if(phi.tbest==0) return 0;
|
|
x=phi.point(phi.tbest);
|
|
f=phi.fbest;
|
|
return phi.tbest;
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> linesearch_minimize(const char *name, linesearch_direction<T> &dir, const OptProblem<T> *o, NRVec<T> &x, const linesearch_hyperparameters<T> &hp, hyperparameters<T> *achieved)
|
|
{
|
|
const int verbose=hp.my_verbose;
|
|
optevaluator<T> e(o,hp.my_maxfunc);
|
|
optstate<T> st(name,hp,e);
|
|
|
|
const int n=x.size();
|
|
NRVec<T> g(n);
|
|
T f=e(x,&g);
|
|
T alpha=0, dphi=0; //of the previous step
|
|
|
|
for(;; ++st.iter)
|
|
{
|
|
if(st.stop(f,g.norm())) break;
|
|
|
|
NRVec<T> p = dir.direction(g);
|
|
bool sd=dir.steepest(); //steepest descent step
|
|
if(p*g >= 0)
|
|
{
|
|
if(dir.indefinite()) {if(verbose) std::cout <<name<<": not a descent direction, steepest descent step\n";}
|
|
else //information spoiled numerically, restart
|
|
{
|
|
if(verbose) std::cout <<name<<": not a descent direction, restart\n";
|
|
dir.reset();
|
|
}
|
|
p = g*(T)-1;
|
|
sd=true;
|
|
}
|
|
|
|
//initial step length: from the previous step, or unit step in x for steepest descent, otherwise the quasi-Newton step
|
|
T dphi_new = p*g;
|
|
T alpha1;
|
|
if(dir.alpha_from_previous() && alpha>0) alpha1=alpha*dphi/dphi_new;
|
|
else alpha1= sd ? 1/p.norm() : (T)1;
|
|
dphi=dphi_new;
|
|
|
|
NRVec<T> xold(x), gold(g);
|
|
T fold=f;
|
|
alpha=linesearch_wolfe(e,x,f,g,p,alpha1,hp);
|
|
if(alpha==0)
|
|
{
|
|
if(e.exhausted()) continue; //reported by stop()
|
|
if(sd) {if(verbose) std::cout <<name<<": line search failed along steepest descent, giving up\n"; break;}
|
|
if(verbose) std::cout <<name<<": line search failed, restart\n";
|
|
dir.reset();
|
|
continue;
|
|
}
|
|
|
|
NRVec<T> s = x-xold;
|
|
NRVec<T> y = g-gold;
|
|
if(verbose>1) std::cout <<name<<" step alpha= "<<alpha<<" |s|= "<<s.norm()<<" s.y= "<<s*y<<std::endl;
|
|
if(!dir.update(s,y,p,g) && verbose) std::cout <<name<<": update skipped\n";
|
|
st.step(f-fold,s.norm());
|
|
}
|
|
|
|
return st.finish(achieved);
|
|
}
|
|
|
|
|
|
//BFGS, DFP, SR1: explicit inverse Hessian approximation
|
|
template<typename T>
|
|
class QN_direction : public linesearch_direction<T> {
|
|
public:
|
|
const BFGS_hyperparameters<T> &hp;
|
|
NRSMat<T> H;
|
|
bool Hinit; //H is the initial guess, rescale it after the first step
|
|
QN_direction(const BFGS_hyperparameters<T> &h, const int n) : hp(h), H(n), Hinit(true) {H=1.;};
|
|
NRVec<T> direction(const NRVec<T> &g) {NRVec<T> p = H*g; p *= (T)-1; return p;}
|
|
bool update(const NRVec<T> &s, const NRVec<T> &y, const NRVec<T> &p, const NRVec<T> &g) {return QN_update_H(H,Hinit,s,y,hp);}
|
|
void reset() {H=1.; Hinit=true;}
|
|
bool steepest() const {return Hinit;}
|
|
bool indefinite() const {return hp.my_update==QN_SR1;}
|
|
};
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> BFGS_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
static const char *names[]={"BFGS","DFP","SR1"};
|
|
BFGS_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("quasi-Newton with line search cannot search for saddle points");
|
|
QN_direction<T> dir(hp,x.size());
|
|
return linesearch_minimize(names[hp.my_update],dir,o,x,hp,achieved);
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> GDIIS_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
const char *name="GDIIS";
|
|
GDIIS_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("GDIIS cannot search for saddle points");
|
|
if(hp.my_diis_dim<1) laerror("GDIIS needs diis_dim >= 1");
|
|
const int verbose=hp.my_verbose;
|
|
optevaluator<T> e(o,hp.my_maxfunc);
|
|
optstate<T> st(name,hp,e);
|
|
const int n=x.size();
|
|
QN_direction<T> dir(hp,n);
|
|
DIIS<NRVec<T>,NRVec<T> > diis(hp.my_diis_dim);
|
|
bool fresh=true; //first step after a (re)start, the quasi-Newton step from the initial Hessian guess has no scale
|
|
NRVec<T> g(n);
|
|
T f=e(x,&g);
|
|
|
|
for(;; ++st.iter)
|
|
{
|
|
if(st.stop(f,g.norm())) break;
|
|
|
|
NRVec<T> p = dir.direction(g); //quasi-Newton step, the error vector of DIIS
|
|
if(p*g >= 0)
|
|
{
|
|
if(verbose) std::cout <<name<<": not a descent direction, restart\n";
|
|
dir.reset();
|
|
diis.reset();
|
|
fresh=true;
|
|
p = g*(T)-1;
|
|
}
|
|
|
|
NRVec<T> xold(x), gold(g);
|
|
T fold=f;
|
|
NRVec<T> s;
|
|
bool accepted=false;
|
|
if(!fresh) //DIIS extrapolation, otherwise a line search is done directly
|
|
{
|
|
s = x+p;
|
|
NRVec<T> err(p);
|
|
diis.extrapolate(s,err,verbose>1,hp.my_diis_eps);
|
|
s -= x;
|
|
if(s*g >= 0 || s.norm() > hp.my_maxsteplength)
|
|
{
|
|
if(verbose) std::cout <<name<<": unreasonable DIIS step, restart\n";
|
|
diis.reset();
|
|
s = p;
|
|
T snorm=s.norm();
|
|
if(snorm > hp.my_maxsteplength) s *= hp.my_maxsteplength/snorm;
|
|
}
|
|
NRVec<T> xnew = x+s;
|
|
NRVec<T> gnew(n);
|
|
T fnew=e(xnew,&gnew);
|
|
if(fnew<f) {x=xnew; f=fnew; g=gnew; accepted=true;}
|
|
else
|
|
{
|
|
if(verbose) std::cout <<name<<": no decrease, restart and line search\n";
|
|
diis.reset();
|
|
}
|
|
}
|
|
if(!accepted) //line search along the quasi-Newton step
|
|
{
|
|
if(e.exhausted()) continue; //reported by stop()
|
|
T alpha=linesearch_wolfe(e,x,f,g,p,dir.steepest() ? 1/p.norm() : (T)1,hp);
|
|
if(alpha==0)
|
|
{
|
|
if(e.exhausted()) continue;
|
|
if(dir.steepest()) {if(verbose) std::cout <<name<<": line search failed along steepest descent, giving up\n"; break;}
|
|
if(verbose) std::cout <<name<<": line search failed, restart\n";
|
|
dir.reset();
|
|
fresh=true;
|
|
continue;
|
|
}
|
|
s = x-xold;
|
|
}
|
|
fresh=false;
|
|
|
|
NRVec<T> y = g-gold;
|
|
if(verbose>1) std::cout <<name<<" step |s|= "<<s.norm()<<" s.y= "<<s*y<<std::endl;
|
|
if(!dir.update(s,y,p,g) && verbose) std::cout <<name<<": update skipped\n";
|
|
st.step(f-fold,s.norm());
|
|
}
|
|
|
|
return st.finish(achieved);
|
|
}
|
|
|
|
|
|
//L-BFGS: last m pairs s = x_{k+1}-x_k, y = g_{k+1}-g_k in a ring buffer
|
|
template<typename T>
|
|
class LBFGS_direction : public linesearch_direction<T> {
|
|
public:
|
|
const LBFGS_hyperparameters<T> &hp;
|
|
int m; //capacity
|
|
int count; //no. of pairs stored
|
|
int newest; //index of the newest pair
|
|
NRVec<NRVec<T> > S,Y;
|
|
NRVec<T> rho; //1/(s.y)
|
|
T gamma; //initial inverse Hessian H0 = gamma I, gamma = s.y/y.y of the newest pair (Nocedal & Wright eq. 7.20)
|
|
LBFGS_direction(const LBFGS_hyperparameters<T> &h) : hp(h), m(h.my_memory), count(0), newest(-1), S(m), Y(m), rho(m), gamma(1) {};
|
|
void push(const NRVec<T> &s, const NRVec<T> &y, const T sy) {newest=(newest+1)%m; S[newest]=s; Y[newest]=y; rho[newest]=1/sy; gamma=sy/(y*y); if(count<m) ++count;}
|
|
NRVec<T> direction(const NRVec<T> &g);
|
|
bool update(const NRVec<T> &s, const NRVec<T> &y, const NRVec<T> &p, const NRVec<T> &g)
|
|
{
|
|
T sy=s*y;
|
|
if(sy <= hp.my_sy_thr*s.norm()*y.norm()) return false; //curvature condition violated numerically
|
|
push(s,y,sy);
|
|
return true;
|
|
}
|
|
void reset() {count=0; newest= -1; gamma=1;}
|
|
bool steepest() const {return count==0;}
|
|
};
|
|
|
|
//returns -H g by the two-loop recursion (Nocedal & Wright alg. 7.4)
|
|
template<typename T>
|
|
NRVec<T> LBFGS_direction<T>::direction(const NRVec<T> &g)
|
|
{
|
|
NRVec<T> q(g);
|
|
NRVec<T> alpha(count);
|
|
int k=newest;
|
|
for(int i=0; i<count; ++i) //newest to oldest
|
|
{
|
|
alpha[i] = rho[k]*(S[k]*q);
|
|
q.axpy(-alpha[i],Y[k]);
|
|
if(i<count-1) k=(k+m-1)%m;
|
|
}
|
|
q *= gamma;
|
|
for(int i=count-1; i>=0; --i) //oldest to newest
|
|
{
|
|
T beta = rho[k]*(Y[k]*q);
|
|
q.axpy(alpha[i]-beta,S[k]);
|
|
k=(k+1)%m;
|
|
}
|
|
q *= (T)-1;
|
|
return q;
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> LBFGS_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
LBFGS_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("L-BFGS cannot search for saddle points");
|
|
if(hp.my_memory<1) laerror("L-BFGS needs memory >= 1");
|
|
LBFGS_direction<T> dir(hp);
|
|
return linesearch_minimize("L-BFGS",dir,o,x,hp,achieved);
|
|
}
|
|
|
|
|
|
//nonlinear CG: p_{k+1} = -g_{k+1} + beta p_k
|
|
template<typename T>
|
|
class CG_direction : public linesearch_direction<T> {
|
|
public:
|
|
const CG_hyperparameters<T> &hp;
|
|
int restart_interval;
|
|
int count; //no. of steps since the last restart
|
|
bool restart; //next direction is steepest descent
|
|
T beta;
|
|
T gg; //g.g of the gradient of the last direction
|
|
NRVec<T> d; //last search direction
|
|
CG_direction(const CG_hyperparameters<T> &h, const int n) : hp(h), restart_interval(h.my_restart>0 ? h.my_restart : n), count(0), restart(true), beta(0), gg(0) {};
|
|
NRVec<T> direction(const NRVec<T> &g) {NRVec<T> p = g*(T)-1; if(!restart) p.axpy(beta,d); gg=g*g; return p;}
|
|
bool update(const NRVec<T> &s, const NRVec<T> &y, const NRVec<T> &p, const NRVec<T> &g);
|
|
void reset() {restart=true; count=0;}
|
|
bool steepest() const {return restart;}
|
|
bool alpha_from_previous() const {return true;}
|
|
};
|
|
|
|
template<typename T>
|
|
bool CG_direction<T>::update(const NRVec<T> &s, const NRVec<T> &y, const NRVec<T> &p, const NRVec<T> &g)
|
|
{
|
|
d=p;
|
|
T gg1=g*g;
|
|
T gy=g*y;
|
|
T dy=p*y;
|
|
restart= ++count>=restart_interval || (hp.my_powell_thr>0 && std::abs(gg1-gy) >= hp.my_powell_thr*gg1); //g_{k+1}.g_k = gg1-gy
|
|
bool curvature= dy>0;
|
|
if(!restart) switch(hp.my_beta)
|
|
{
|
|
case CG_FR: beta=gg1/gg; break;
|
|
case CG_PR: beta=std::max((T)0,gy/gg); break;
|
|
case CG_HS: if(curvature) beta=std::max((T)0,gy/dy); else restart=true; break;
|
|
case CG_DY: if(curvature) beta=gg1/dy; else restart=true; break;
|
|
case CG_HZ: if(curvature) beta=std::max((gy-2*(p*g)*(y*y)/dy)/dy, -1/(p.norm()*std::min((T)0.01,std::sqrt(gg)))); else restart=true; break;
|
|
default: laerror("unknown CG beta formula");
|
|
}
|
|
if(restart)
|
|
{
|
|
count=0;
|
|
if(hp.my_verbose>1) std::cout <<"CG restart\n";
|
|
}
|
|
return true;
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> CG_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
static const char *names[]={"CG-FR","CG-PR+","CG-HS+","CG-DY","CG-HZ"};
|
|
CG_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("CG cannot search for saddle points");
|
|
CG_direction<T> dir(hp,x.size());
|
|
return linesearch_minimize(names[hp.my_beta],dir,o,x,hp,achieved);
|
|
}
|
|
|
|
|
|
|
|
//simplex of n+1 vertices with their function values
|
|
template<typename T>
|
|
class NM_simplex {
|
|
public:
|
|
int n;
|
|
NRVec<NRVec<T> > v; //vertices
|
|
NRVec<T> f; //function values
|
|
NRVec<T> sum; //sum of the vertices
|
|
int best, second, worst; //indices of the lowest, second highest and highest function value
|
|
NM_simplex(const NRVec<T> &x0, const T h, optevaluator<T> &e);
|
|
void order();
|
|
void sumup() {sum=v[0]; for(int i=1; i<=n; ++i) sum+=v[i];}
|
|
void replace(const int i, const NRVec<T> &x, const T fx) {sum-=v[i]; sum+=x; v[i]=x; f[i]=fx;}
|
|
NRVec<T> centroid() const {NRVec<T> c(sum); c-=v[worst]; c/=(T)n; return c;} //of all vertices except the worst one
|
|
NRVec<T> point(const NRVec<T> &c, const T t) const {NRVec<T> x = c*(1+t); x.axpy(-t,v[worst]); return x;} //c + t*(c - x_worst)
|
|
T diameter() const;
|
|
void shrink(const T sigma, optevaluator<T> &e);
|
|
};
|
|
|
|
//initial simplex x0, x0 + h e_i
|
|
template<typename T>
|
|
NM_simplex<T>::NM_simplex(const NRVec<T> &x0, const T h, optevaluator<T> &e) : n(x0.size()), v(n+1), f(n+1), best(0), second(0), worst(0)
|
|
{
|
|
v[0]=x0;
|
|
f[0]=e(x0);
|
|
for(int i=1; i<=n; ++i)
|
|
{
|
|
v[i]=x0;
|
|
v[i].copyonwrite();
|
|
v[i][i-1] += h;
|
|
f[i]=e(v[i]);
|
|
}
|
|
sumup();
|
|
}
|
|
|
|
template<typename T>
|
|
void NM_simplex<T>::order()
|
|
{
|
|
best=worst=0;
|
|
for(int i=1; i<=n; ++i)
|
|
{
|
|
if(f[i]<f[best]) best=i;
|
|
if(f[i]>=f[worst]) worst=i;
|
|
}
|
|
second=best;
|
|
for(int i=0; i<=n; ++i) if(i!=worst && f[i]>=f[second]) second=i;
|
|
}
|
|
|
|
template<typename T>
|
|
T NM_simplex<T>::diameter() const
|
|
{
|
|
T d=0;
|
|
for(int i=0; i<=n; ++i) if(i!=best)
|
|
{
|
|
NRVec<T> dx = v[i]-v[best];
|
|
d=std::max(d,dx.norm());
|
|
}
|
|
return d;
|
|
}
|
|
|
|
//all vertices move towards the best one, as far as the evaluation budget allows
|
|
template<typename T>
|
|
void NM_simplex<T>::shrink(const T sigma, optevaluator<T> &e)
|
|
{
|
|
for(int i=0; i<=n; ++i) if(i!=best)
|
|
{
|
|
if(e.exhausted()) break;
|
|
NRVec<T> x = v[best]*(1-sigma);
|
|
x.axpy(sigma,v[i]);
|
|
v[i]=x;
|
|
f[i]=e(x);
|
|
}
|
|
sumup();
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> NelderMead_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
const char *name="Nelder-Mead";
|
|
NelderMead_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("Nelder-Mead cannot search for saddle points");
|
|
const int verbose=hp.my_verbose;
|
|
const int n=x.size();
|
|
T alpha=hp.my_alpha, gamma=hp.my_gamma, rho=hp.my_rho, sigma=hp.my_sigma;
|
|
if(hp.my_adaptive) {alpha=1; gamma=1+(T)2/n; rho=0.75-(T)0.5/n; sigma=1-(T)1/n;}
|
|
optevaluator<T> e(o,hp.my_maxfunc);
|
|
optstate<T> st(name,hp,e);
|
|
NM_simplex<T> s(x,hp.my_initstep,e);
|
|
const T huge=std::numeric_limits<T>::max();
|
|
|
|
for(;; ++st.iter)
|
|
{
|
|
s.order();
|
|
T spread=s.f[s.worst]-s.f[s.best];
|
|
st.step(spread, spread<hp.my_f_thr ? s.diameter() : huge); //diameter needed only if the spread is small
|
|
if(st.stop(s.f[s.best])) break;
|
|
if(st.iter%(n+1)==0) s.sumup(); //avoid accumulation of rounding errors
|
|
|
|
const char *op;
|
|
NRVec<T> c=s.centroid();
|
|
NRVec<T> xr=s.point(c,alpha);
|
|
T fr=e(xr);
|
|
if(fr<s.f[s.best]) //try expansion
|
|
{
|
|
if(e.exhausted()) {s.replace(s.worst,xr,fr); continue;}
|
|
NRVec<T> xe=s.point(c,alpha*gamma);
|
|
T fe=e(xe);
|
|
if(fe<fr) {s.replace(s.worst,xe,fe); op="expansion";}
|
|
else {s.replace(s.worst,xr,fr); op="reflection";}
|
|
}
|
|
else if(fr<s.f[s.second]) {s.replace(s.worst,xr,fr); op="reflection";}
|
|
else //contraction, outside if the reflected point is better than the worst one
|
|
{
|
|
bool outside= fr<s.f[s.worst];
|
|
if(e.exhausted()) {if(outside) s.replace(s.worst,xr,fr); continue;}
|
|
NRVec<T> xc=s.point(c, outside ? alpha*rho : -rho);
|
|
T fc=e(xc);
|
|
if(outside ? fc<=fr : fc<s.f[s.worst]) {s.replace(s.worst,xc,fc); op= outside ? "outside contraction" : "inside contraction";}
|
|
else {s.shrink(sigma,e); op="shrink";}
|
|
}
|
|
if(verbose>1) std::cout <<name<<" "<<op<<std::endl;
|
|
}
|
|
|
|
s.order();
|
|
x=s.v[s.best];
|
|
st.step(s.f[s.worst]-s.f[s.best],s.diameter()); //exact values for achieved
|
|
return st.finish(achieved);
|
|
}
|
|
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> Powell_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
const char *name="Powell";
|
|
Powell_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("Powell cannot search for saddle points");
|
|
const int verbose=hp.my_verbose;
|
|
const T abstol=0.01*hp.my_x_thr; //line minima have to be more precise than the requested accuracy in x
|
|
const int n=x.size();
|
|
optevaluator<T> e(o,hp.my_maxfunc);
|
|
optstate<T> st(name,hp,e);
|
|
T f=e(x);
|
|
|
|
NRVec<NRVec<T> > u(n); //normalized directions
|
|
for(int i=0; i<n; ++i) {u[i]=NRVec<T>((T)0,n); u[i][i]=1;}
|
|
NRVec<T> tstep(hp.my_initstep,n); //trial steps of the line minimizations, from the last step along each direction
|
|
|
|
for(;; ++st.iter)
|
|
{
|
|
if(st.stop(f)) break;
|
|
NRVec<T> x0(x);
|
|
T f0=f;
|
|
T delta=0; //largest decrease along a single direction
|
|
int ibig=0;
|
|
for(int i=0; i<n && !e.exhausted(); ++i)
|
|
{
|
|
T fprev=f;
|
|
T t=linemin_brent(e,x,f,u[i],tstep[i],hp.my_linmin_tol,abstol,hp.my_maxlinemin,verbose);
|
|
if(t!=0) tstep[i]=std::abs(t);
|
|
if(fprev-f>delta) {delta=fprev-f; ibig=i;}
|
|
}
|
|
|
|
//new direction x-x0 replaces the one of the largest decrease, if Powell's test indicates it is worth
|
|
NRVec<T> unew = x-x0;
|
|
T unorm=unew.norm();
|
|
if(unorm>0 && !e.exhausted())
|
|
{
|
|
NRVec<T> xe = x+unew; //extrapolated point
|
|
T fe=e(xe);
|
|
T a=f0-f-delta;
|
|
T b=f0-fe;
|
|
if(fe<f0 && 2*(f0-2*f+fe)*a*a < delta*b*b)
|
|
{
|
|
if(verbose>1) std::cout <<name<<": direction "<<ibig<<" replaced\n";
|
|
unew /= unorm;
|
|
u[ibig]=u[n-1]; tstep[ibig]=tstep[n-1];
|
|
u[n-1]=unew; tstep[n-1]=unorm;
|
|
if(!e.exhausted()) linemin_brent(e,x,f,unew,unorm,hp.my_linmin_tol,abstol,hp.my_maxlinemin,verbose);
|
|
}
|
|
}
|
|
NRVec<T> dx = x-x0;
|
|
st.step(f0-f,dx.norm());
|
|
}
|
|
|
|
return st.finish(achieved);
|
|
}
|
|
|
|
|
|
|
|
//natural gradient (G + reg I)^-1 g from the gradient of an inner estimator and the metric G of the problem
|
|
template<typename T>
|
|
class natural_gradient_estimator : public gradient_estimator<T> {
|
|
public:
|
|
gradient_estimator<T> &inner;
|
|
const OptMetric<T> *m;
|
|
T reg;
|
|
natural_gradient_estimator(gradient_estimator<T> &in, const OptMetric<T> *mm, const T r) : inner(in), m(mm), reg(r) {};
|
|
T operator()(optevaluator<T> &e, const NRVec<T> &x, NRVec<T> &g, const int k)
|
|
{
|
|
T f=inner(e,x,g,k);
|
|
NRSMat<T> G(x.size());
|
|
m->metric(x,G);
|
|
if(reg!=0) G+=reg;
|
|
linear_solve(G,g);
|
|
return f;
|
|
}
|
|
int cost() const {return inner.cost();}
|
|
bool exact() const {return inner.exact();}
|
|
T calibrate(optevaluator<T> &e, const NRVec<T> &x, const firstorder_hyperparameters<T> &hp) {return inner.calibrate(e,x,hp);}
|
|
};
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> firstorder_minimize(const char *name, firstorder_step<T> &stepper, const OptProblem<T> *o, NRVec<T> &x, const firstorder_hyperparameters<T> &hp, hyperparameters<T> *achieved, gradient_estimator<T> *est)
|
|
{
|
|
gradient_estimator<T> exact;
|
|
gradient_estimator<T> *plain = est ? est : &exact;
|
|
const OptMetric<T> *metric=NULL;
|
|
if(hp.my_natural)
|
|
{
|
|
metric=optcapability<OptMetric<T> >(o);
|
|
if(!metric) laerror("natural gradient needs a problem providing the metric (OptMetric)");
|
|
}
|
|
natural_gradient_estimator<T> natural(*plain,metric,hp.my_metric_reg);
|
|
gradient_estimator<T> &grad = hp.my_natural ? natural : *plain;
|
|
optevaluator<T> e(o,hp.my_maxfunc);
|
|
optstate<T> st(name,hp,e);
|
|
const int n=x.size();
|
|
const T lr0=grad.calibrate(e,x,hp);
|
|
if(hp.my_verbose && lr0!=hp.my_lr) std::cout <<name<<": calibrated learning rate "<<lr0<<std::endl;
|
|
NRVec<T> g(n);
|
|
T f=grad(e,x,g,0);
|
|
|
|
for(;; ++st.iter)
|
|
{
|
|
if(st.stop(f,grad.exact() ? g.norm() : (T)-1,grad.cost())) break;
|
|
T lr=lr0/std::pow(1+hp.my_lr_decay*st.iter,hp.my_lr_power);
|
|
NRVec<T> dx = stepper.step(g,st.iter+1);
|
|
dx *= -lr;
|
|
if(hp.my_weight_decay!=0) dx.axpy(-lr*hp.my_weight_decay,x);
|
|
x += dx;
|
|
T fold=f;
|
|
NRVec<T> gnew(n); //fresh vector, the stepper may keep the previous gradient
|
|
f=grad(e,x,gnew,st.iter+1);
|
|
g=gnew;
|
|
if(grad.exact()) st.step(f-fold,dx.norm()); //for estimates, small changes are no sign of convergence
|
|
}
|
|
|
|
if(hp.my_return_best && e.fbest<f)
|
|
{
|
|
if(hp.my_verbose) std::cout <<name<<": returning the best point found, f= "<<e.fbest<<std::endl;
|
|
x=e.xbest;
|
|
st.f=e.fbest;
|
|
}
|
|
optresult<T> r=st.finish(achieved);
|
|
firstorder_hyperparameters<T> *fach=dynamic_cast<firstorder_hyperparameters<T> *>(achieved);
|
|
if(fach) fach->my_lr=lr0;
|
|
return r;
|
|
}
|
|
|
|
|
|
//SGD with momentum
|
|
template<typename T>
|
|
class SGD_stepper : public firstorder_step<T> {
|
|
public:
|
|
const SGD_hyperparameters<T> &hp;
|
|
NRVec<T> v; //velocity
|
|
SGD_stepper(const SGD_hyperparameters<T> &h) : hp(h) {};
|
|
NRVec<T> step(const NRVec<T> &g, const int k)
|
|
{
|
|
if(k==1) v=g;
|
|
else {v *= hp.my_momentum; v += g;}
|
|
if(!hp.my_nesterov) return v;
|
|
NRVec<T> d(g);
|
|
d.axpy(hp.my_momentum,v);
|
|
return d;
|
|
}
|
|
};
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> QNG_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
QNG_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
hp.my_natural=true; //also if the caller passed e.g. SGD_hyperparameters
|
|
if(hp.my_saddle_order) laerror("QNG cannot search for saddle points");
|
|
SGD_stepper<T> stepper(hp);
|
|
return firstorder_minimize("QNG",stepper,o,x,hp,achieved);
|
|
}
|
|
|
|
|
|
//approximate solution of min c.z subject to A z >= r and |(z_0,...,z_{nd-1})| <= delta, nd = nz or nz-1, from a feasible z by an active-set method:
|
|
//z moves along the projection of -c orthogonal to the normals of the active constraints until a further constraint becomes active,
|
|
//the trust region boundary is reached (which finishes the calculation as in COBYLA), or the multipliers show optimality;
|
|
//the projection is computed by least squares, which tolerates dependent normals at degenerate points
|
|
template<typename T>
|
|
static void lp_trustregion(const NRVec<T> &c, const NRMat<T> &A, const NRVec<T> &r, const int nd, const T delta, NRVec<T> &z)
|
|
{
|
|
const int nz=c.size();
|
|
const int m=r.size();
|
|
const T tiny=1e3*std::numeric_limits<T>::epsilon()*c.norm();
|
|
NRVec<int> active(m);
|
|
active.clear();
|
|
NRVec<int> cols(nz);
|
|
for(int j=0; j<nz; ++j) cols[j]=j;
|
|
const NRVec<T> &zc=z; //element access for reading only
|
|
const int maxit=2*(m+nz)+10;
|
|
for(int it=0; it<maxit; ++it)
|
|
{
|
|
int k=0;
|
|
for(int i=0; i<m; ++i) if(active[i]) ++k;
|
|
NRVec<int> rows(k);
|
|
k=0;
|
|
for(int i=0; i<m; ++i) if(active[i]) rows[k++]=i;
|
|
NRVec<T> s = c*(T)-1; //search direction
|
|
NRVec<T> lambda; //multipliers of the active constraints
|
|
if(k>0)
|
|
{
|
|
NRMat<T> N=A.submatrix(rows,cols); //normals of the active constraints
|
|
NRMat<T> Nt=N.transpose();
|
|
lambda=svd_solve(Nt,c); //least squares c ~ N^T lambda
|
|
NRVec<T> nl=lambda*N;
|
|
s+=nl;
|
|
}
|
|
if(s.norm()<=tiny) //stationary on the active set, optimal if the multipliers are nonnegative
|
|
{
|
|
int drop= -1;
|
|
T lmin=0;
|
|
for(int j=0; j<k; ++j) if(lambda[j]<lmin) {lmin=lambda[j]; drop=j;}
|
|
if(drop<0) return;
|
|
active[rows[drop]]=0;
|
|
continue;
|
|
}
|
|
//step to the first blocking constraint
|
|
NRVec<T> Az=A*z;
|
|
NRVec<T> As=A*s;
|
|
T alpha=std::numeric_limits<T>::max();
|
|
int block= -1;
|
|
for(int i=0; i<m; ++i) if(!active[i] && As[i]<0)
|
|
{
|
|
T a=std::max((T)0,(Az[i]-r[i])/(-As[i]));
|
|
if(a<alpha) {alpha=a; block=i;}
|
|
}
|
|
//step to the trust region boundary |zd + a sd| = delta
|
|
T zz=z*z, ss=s*s, zs=z*s;
|
|
if(nd<nz) {zz-=zc[nz-1]*zc[nz-1]; ss-=s[nz-1]*s[nz-1]; zs-=zc[nz-1]*s[nz-1];}
|
|
if(ss>0)
|
|
{
|
|
T ab=(-zs+std::sqrt(std::max((T)0,zs*zs-ss*(zz-delta*delta))))/ss;
|
|
if(ab<alpha) {z.axpy(ab,s); return;}
|
|
}
|
|
if(block<0) return;
|
|
z.axpy(alpha,s);
|
|
active[block]=1;
|
|
}
|
|
}
|
|
|
|
|
|
//COBYLA trust region step: minimizes g.d subject to the linearized constraints c + A d >= 0 within |d| <= rho; if they cannot be satisfied
|
|
//in the trust region, the largest violation t is minimized first (stage 1) and then kept, c + A d >= -t (stage 2) (Powell 1994)
|
|
template<typename T>
|
|
static NRVec<T> cobyla_step(const NRVec<T> &g, const NRMat<T> &A, const NRVec<T> &c, const T rho)
|
|
{
|
|
const int n=g.size();
|
|
const int m=c.size();
|
|
NRVec<T> d((T)0,n);
|
|
if(m==0) {T gn=g.norm(); if(gn>0) d.axpy(-rho/gn,g); return d;}
|
|
T t=std::max((T)0,-c.min());
|
|
if(t>0) //stage 1 in the variables (d,t): min t subject to c_i + a_i.d + t >= 0, t >= 0
|
|
{
|
|
NRMat<T> A1(m+1,n+1);
|
|
A1.clear();
|
|
A1.storesubmatrix(0,0,A);
|
|
for(int i=0; i<=m; ++i) A1(i,n)=1;
|
|
NRVec<T> r1(m+1); //r1_i = -c_i, r1_m = 0
|
|
r1.clear();
|
|
for(int i=0; i<m; ++i) r1[i]= -c[i];
|
|
NRVec<T> c1((T)0,n+1);
|
|
c1[n]=1;
|
|
NRVec<T> z((T)0,n+1);
|
|
z[n]=t;
|
|
lp_trustregion(c1,A1,r1,n,rho,z);
|
|
t=z[n];
|
|
for(int j=0; j<n; ++j) d[j]=z[j];
|
|
}
|
|
//stage 2: min g.d subject to c_i + a_i.d >= -t
|
|
NRVec<T> r2 = c*(T)-1;
|
|
r2-=t;
|
|
lp_trustregion(g,A,r2,n,rho,d);
|
|
return d;
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
static T cobyla_violation(const NRVec<T> &c) {return c.size()>0 ? std::max((T)0,-c.min()) : 0;}
|
|
|
|
//f and the constraint values at x
|
|
template<typename T>
|
|
static T cobyla_eval(optevaluator<T> &e, const OptConstraints<T> *con, const NRVec<T> &x, NRVec<T> &c)
|
|
{
|
|
T f=e(x);
|
|
if(con) con->constraints(x,c);
|
|
return f;
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> Cobyla_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
const char *name="COBYLA";
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|
Cobyla_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("COBYLA cannot search for saddle points");
|
|
const T acc_sigma=0.25, acc_eta=2.1, geo_gamma=0.5; //simplex acceptability and geometry step (Powell 1994)
|
|
const int verbose=hp.my_verbose;
|
|
const int n=x.size();
|
|
const OptConstraints<T> *con=optcapability<OptConstraints<T> >(o);
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|
const int m= con ? con->nconstraints() : 0;
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|
optevaluator<T> e(o,hp.my_maxfunc);
|
|
optstate<T> st(name,hp,e);
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|
|
|
//initial simplex x, x + rhobeg e_j with function and constraint values
|
|
T rho=hp.my_rhobeg; //resolution, lower bound of the trust region radius
|
|
T delta=rho; //trust region radius, always equal to rho in Powell's original COBYLA
|
|
const bool flexible=hp.my_flexible_radius;
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|
T mu=0; //penalty parameter of the merit function f + mu*(largest constraint violation)
|
|
NRVec<NRVec<T> > v(n+1), cv(n+1);
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|
NRVec<T> fv(n+1);
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|
for(int j=0; j<=n; ++j)
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|
{
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|
v[j]=x;
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|
if(j>0) {v[j].copyonwrite(); v[j][j-1]+=rho;}
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|
cv[j]=NRVec<T>(m);
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|
fv[j]=cobyla_eval(e,con,v[j],cv[j]);
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|
}
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|
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int best=0;
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|
T fprev=fv[0];
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NRVec<int> other(n); //vertices other than the best one
|
|
bool lastsuccess=false; //the last trust region step was successful, the next one follows without checking the simplex
|
|
for(;; ++st.iter)
|
|
{
|
|
best=0;
|
|
for(int j=1; j<=n; ++j) if(fv[j]+mu*cobyla_violation(cv[j]) < fv[best]+mu*cobyla_violation(cv[best])) best=j;
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|
if(st.iter>0) {st.step(fprev-fv[best],rho); fprev=fv[best];}
|
|
if(st.stop(fv[best])) break;
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|
for(int j=0, k=0; j<=n; ++j) if(j!=best) other[k++]=j;
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|
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|
//displacements d_k of the other vertices from the best one and the dual basis B = D^-T, b_k.d_l = delta_kl
|
|
NRMat<T> D(n,n);
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|
for(int k=0; k<n; ++k) {NRVec<T> dk=v[other[k]]-v[best]; D.rowset(dk,k);}
|
|
NRMat<T> B=D.inverse().transpose();
|
|
//linear models: gradient g of f, gradients of the constraints in the rows of A, values c0 at the best vertex
|
|
NRVec<T> df(n);
|
|
for(int k=0; k<n; ++k) df[k]=fv[other[k]]-fv[best];
|
|
NRVec<T> g=df*B;
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|
const NRVec<T> c0(cv[best]);
|
|
NRMat<T> A;
|
|
if(m>0)
|
|
{
|
|
NRMat<T> dC(n,m);
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|
for(int k=0; k<n; ++k) {NRVec<T> dc=cv[other[k]]-c0; dC.rowset(dc,k);}
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|
A=dC.transpose()*B;
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|
}
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|
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|
//acceptability of the simplex: distances eta_k = |d_k| <= acc_eta delta and sigma_k = 1/|b_k| >= acc_sigma rho of the vertices from the opposite faces
|
|
int worst= -1;
|
|
T etamax=acc_eta*delta;
|
|
for(int k=0; k<n; ++k) {T eta=D.row(k).norm(); if(eta>etamax) {etamax=eta; worst=k;}}
|
|
if(worst<0)
|
|
{
|
|
T sigmin=acc_sigma*rho;
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|
for(int k=0; k<n; ++k) {T sigma=1/B.row(k).norm(); if(sigma<sigmin) {sigmin=sigma; worst=k;}}
|
|
}
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|
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|
if(!lastsuccess && worst>=0) //geometry step: the worst vertex moves to the best one + a step along the normal of its opposite face, with the sign preferred by the merit model
|
|
{
|
|
NRVec<T> dir=B.row(worst);
|
|
//step length geo_gamma rho, or with flexible radius max(min(0.1 DIST, 0.5 delta), rho) (NEWUOA eq. (6.16)), DIST the distance of the replaced vertex
|
|
T len= flexible ? std::max(std::min((T)0.1*D.row(worst).norm(),(T)0.5*delta),rho) : geo_gamma*rho;
|
|
dir*= len/dir.norm();
|
|
T mplus=g*dir, mminus= -mplus;
|
|
if(m>0)
|
|
{
|
|
NRVec<T> ad=A*dir;
|
|
NRVec<T> cp(c0), cm(c0);
|
|
cp+=ad; cm-=ad;
|
|
mplus+=mu*cobyla_violation(cp);
|
|
mminus+=mu*cobyla_violation(cm);
|
|
}
|
|
if(mminus<mplus) dir*= (T)-1;
|
|
if(verbose>1) std::cout <<name<<": geometry step of vertex "<<other[worst]<<std::endl;
|
|
NRVec<T> xn=v[best]+dir;
|
|
NRVec<T> cn(m);
|
|
T fn=cobyla_eval(e,con,xn,cn);
|
|
v[other[worst]]=xn; fv[other[worst]]=fn; cv[other[worst]]=cn;
|
|
continue;
|
|
}
|
|
|
|
//trust region step
|
|
lastsuccess=false;
|
|
NRVec<T> d=cobyla_step(g,A,c0,delta);
|
|
const T dnorm=d.norm();
|
|
const T deltaold=delta; //trust region radius of this step
|
|
T ratio= -1; //reduction ratio of the merit function
|
|
const bool shortstep= dnorm<0.5*rho;
|
|
bool failed=shortstep;
|
|
if(shortstep && flexible) delta=std::max((T)0.1*delta,rho); //NEWUOA Box 15
|
|
if(!failed)
|
|
{
|
|
const T v0=cobyla_violation(c0);
|
|
NRVec<T> cd(c0);
|
|
if(m>0) {NRVec<T> ad=A*d; cd+=ad;}
|
|
T fpred= -(g*d); //predicted reductions of f and of the violation
|
|
T vpred=v0-cobyla_violation(cd);
|
|
if(vpred>0 && mu < -1.5*fpred/vpred) //penalty has to make the predicted reduction of the merit function positive
|
|
{
|
|
mu= -2*fpred/vpred;
|
|
if(verbose>1) std::cout <<name<<": penalty parameter "<<mu<<std::endl;
|
|
int b2=best;
|
|
for(int j=0; j<=n; ++j) if(fv[j]+mu*cobyla_violation(cv[j]) < fv[b2]+mu*cobyla_violation(cv[b2])) b2=j;
|
|
if(b2!=best) continue; //another vertex is best with the new penalty
|
|
}
|
|
T pred=fpred+mu*vpred;
|
|
if(pred<=0) failed=true;
|
|
else
|
|
{
|
|
NRVec<T> xn=v[best]+d;
|
|
NRVec<T> cn(m);
|
|
T fn=cobyla_eval(e,con,xn,cn);
|
|
T phi0=fv[best]+mu*v0;
|
|
T phin=fn+mu*cobyla_violation(cn);
|
|
ratio=(phi0-phin)/pred;
|
|
if(verbose>1) std::cout <<name<<" step |d|= "<<dnorm<<" delta= "<<delta<<" rho= "<<rho<<" f= "<<fn<<" violation= "<<cobyla_violation(cn)<<" ratio= "<<ratio<<std::endl;
|
|
//vertex replaced by the new point (Powell 1994): the one giving the largest simplex volume, i.e. the largest barycentric coordinate |l_k|
|
|
//(without improvement only if the volume increases), but preferably the most distant one beyond 1.1 rho, if the new point
|
|
//keeps an acceptable distance |l_k| sigma_k from its opposite face; the best vertex is kept, a better new point becomes the best one
|
|
const bool improved= phin<phi0;
|
|
NRVec<T> l=B*d;
|
|
int rep= -1;
|
|
T lmax= improved ? 0 : 1;
|
|
for(int k=0; k<n; ++k) if(std::abs(l[k])>lmax) {lmax=std::abs(l[k]); rep=k;}
|
|
T edgmax=1.1*delta;
|
|
for(int k=0; k<n; ++k)
|
|
{
|
|
T sigma=1/B.row(k).norm();
|
|
T sigbar=std::abs(l[k])*sigma;
|
|
if(sigbar>=acc_sigma*rho || sigbar>=sigma)
|
|
{
|
|
T dist;
|
|
if(improved) {NRVec<T> dk=v[other[k]]-xn; dist=dk.norm();}
|
|
else dist=D.row(k).norm();
|
|
if(dist>edgmax) {edgmax=dist; rep=k;}
|
|
}
|
|
}
|
|
if(rep>=0) {v[other[rep]]=xn; fv[other[rep]]=fn; cv[other[rep]]=cn;}
|
|
}
|
|
}
|
|
if(flexible && !shortstep) //revision of the trust region radius by the reduction ratio (NEWUOA eq. (7.1))
|
|
{
|
|
T dint;
|
|
if(ratio<=0.1) dint=0.5*dnorm;
|
|
else if(ratio<=0.7) dint=std::max(dnorm,(T)0.5*delta);
|
|
else dint=std::max(2*dnorm,(T)0.5*delta);
|
|
delta= dint<=1.5*rho ? rho : dint;
|
|
}
|
|
if(!failed && ratio>=0.1) {lastsuccess=true; continue;}
|
|
|
|
//failed trust region step: improve the simplex at the next iteration, or reduce rho if it is acceptable
|
|
if(worst>=0) continue;
|
|
//the work with the current rho continues if the step was not restricted by rho or improved the point (NEWUOA Box 10, where |d| > rho is tested;
|
|
//here the radius of the step is compared instead, as |d| = delta = rho can be slightly above rho due to rounding)
|
|
if(flexible && ((!shortstep && deltaold>rho) || delta>rho || ratio>0)) continue;
|
|
if(rho<=hp.my_rhoend) {st.converged=true; break;}
|
|
if(flexible) //NEWUOA eq. (7.6) and Box 12
|
|
{
|
|
T rhonew;
|
|
if(rho<=16*hp.my_rhoend) rhonew=hp.my_rhoend;
|
|
else if(rho<=250*hp.my_rhoend) rhonew=std::sqrt(rho*hp.my_rhoend);
|
|
else rhonew=0.1*rho;
|
|
delta=std::max((T)0.5*rho,rhonew);
|
|
rho=rhonew;
|
|
}
|
|
else
|
|
{
|
|
rho*=0.5;
|
|
if(rho<=1.5*hp.my_rhoend) rho=hp.my_rhoend;
|
|
delta=rho;
|
|
}
|
|
if(verbose>1) std::cout <<name<<": rho= "<<rho<<" delta= "<<delta<<std::endl;
|
|
}
|
|
|
|
best=0;
|
|
for(int j=1; j<=n; ++j) if(fv[j]+mu*cobyla_violation(cv[j]) < fv[best]+mu*cobyla_violation(cv[best])) best=j;
|
|
x=v[best];
|
|
st.f=fv[best];
|
|
if(verbose && m>0) std::cout <<name<<": largest constraint violation "<<cobyla_violation(cv[best])<<std::endl;
|
|
return st.finish(achieved);
|
|
}
|
|
|
|
|
|
//trigonometric polynomial a_0 + sum_{k=1}^R (a_k cos(k w t) + b_k sin(k w t)) of the single variable t = x[0]
|
|
template<typename T>
|
|
class trigpolynomial : public OptProblem<T> {
|
|
public:
|
|
NRVec<T> a, b;
|
|
T w;
|
|
trigpolynomial(const int R, const T ww) : a(R+1), b(R+1), w(ww) {};
|
|
T funcgrad(const NRVec<T> &x, NRVec<T> *grad) const
|
|
{
|
|
if(grad) laerror("trigpolynomial provides only the function");
|
|
T f=a[0];
|
|
for(int k=1; k<a.size(); ++k) f+= a[k]*std::cos(k*w*x[0]) + b[k]*std::sin(k*w*x[0]);
|
|
return f;
|
|
}
|
|
};
|
|
|
|
//exact minimization of f along x_i, a trigonometric polynomial of degree R with the given period: reconstruction from
|
|
//2R+1 equidistant samples t_j = j*period/(2R+1) (the one at t=0 is the current f, 2R new evaluations) by discrete Fourier transform
|
|
//and minimization of the reconstruction; returns false if the evaluation budget is insufficient (x,f then unchanged)
|
|
template<typename T>
|
|
static bool trig_minimize(optevaluator<T> &e, NRVec<T> &x, T &f, const int i, const int R, const T period, const bool reevaluate, const int verbose)
|
|
{
|
|
const int N=2*R+1;
|
|
if(e.remaining()<N-1+(reevaluate?1:0)) return false;
|
|
const T w=2*M_PI/period;
|
|
const T xi=((const NRVec<T> &)x)[i]; //const access, x may be shared with the evaluator
|
|
NRVec<T> fs(N); //samples
|
|
fs[0]=f;
|
|
NRVec<T> xs(x);
|
|
for(int j=1; j<N; ++j)
|
|
{
|
|
xs.copyonwrite(); //the evaluator may keep a reference to the previous sample as its best point
|
|
xs[i]=xi+j*period/N;
|
|
fs[j]=e(xs);
|
|
}
|
|
|
|
trigpolynomial<T> p(R,w);
|
|
p.a[0]=fs.sum()/N;
|
|
for(int k=1; k<=R; ++k)
|
|
{
|
|
p.a[k]=p.b[k]=0;
|
|
for(int j=0; j<N; ++j) {T phi=2*M_PI*k*j/N; p.a[k]+= fs[j]*std::cos(phi); p.b[k]+= fs[j]*std::sin(phi);}
|
|
p.a[k]*= (T)2/N; p.b[k]*= (T)2/N;
|
|
}
|
|
|
|
T t, fmin;
|
|
if(R==1) //closed form: a_1 cos(wt) + b_1 sin(wt) = A cos(wt - phi), minimum at wt = phi + pi
|
|
{
|
|
t=(std::atan2(p.b[1],p.a[1])+M_PI)/w;
|
|
fmin=p.a[0]-std::sqrt(p.a[1]*p.a[1]+p.b[1]*p.b[1]);
|
|
}
|
|
else //global minimum of the reconstruction: best of a dense grid, refined by Brent's method (no evaluations of the problem)
|
|
{
|
|
const int M=8*N;
|
|
NRVec<T> tt(1);
|
|
int jbest=0;
|
|
T fbest=0;
|
|
for(int j=0; j<M; ++j) {tt[0]=j*period/M; T ft=p.funcgrad(tt,NULL); if(j==0 || ft<fbest) {fbest=ft; jbest=j;}}
|
|
optevaluator<T> pe(&p);
|
|
tt[0]=jbest*period/M;
|
|
NRVec<T> u((T)1,1);
|
|
linemin_brent(pe,tt,fbest,u,period/M,(T)1e-10,(T)1e-12*period,100);
|
|
t=tt[0];
|
|
fmin=fbest;
|
|
}
|
|
t-= period*std::floor(t/period+0.5); //shortest step, t in (-period/2,period/2]
|
|
x.copyonwrite();
|
|
x[i]=xi+t;
|
|
f= reevaluate ? e(x) : fmin;
|
|
if(verbose>1) std::cout <<"trigonometric minimization of parameter "<<i<<" degree "<<R<<" step "<<t<<" f= "<<f<<std::endl;
|
|
return true;
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> Rotosolve_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
const char *name="Rotosolve";
|
|
Rotosolve_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("Rotosolve cannot search for saddle points");
|
|
const int verbose=hp.my_verbose;
|
|
const T abstol=0.01*hp.my_x_thr; //as in Powell's method
|
|
const int n=x.size();
|
|
const OptTrigonometric<T> *trig=optcapability<OptTrigonometric<T> >(o);
|
|
if(!trig && verbose) std::cout <<name<<": the problem is not trigonometric, line minimizations only\n";
|
|
optevaluator<T> e(o,hp.my_maxfunc);
|
|
optstate<T> st(name,hp,e);
|
|
T f=e(x);
|
|
NRVec<T> tstep(hp.my_initstep,n); //trial steps of the line minimizations
|
|
|
|
for(;; ++st.iter)
|
|
{
|
|
if(st.stop(f)) break;
|
|
NRVec<T> x0(x);
|
|
T f0=f;
|
|
bool budget=true;
|
|
for(int i=0; i<n && budget; ++i)
|
|
{
|
|
const int R= trig ? trig->degree(i) : 0;
|
|
if(R>0) budget=trig_minimize(e,x,f,i,R,trig->period(i),hp.my_reevaluate,verbose);
|
|
else
|
|
{
|
|
NRVec<T> u((T)0,n);
|
|
u[i]=1;
|
|
T t=linemin_brent(e,x,f,u,tstep[i],hp.my_linmin_tol,abstol,hp.my_maxlinemin,verbose);
|
|
if(t!=0) tstep[i]=std::abs(t);
|
|
budget=!e.exhausted();
|
|
}
|
|
}
|
|
NRVec<T> dx = x-x0;
|
|
st.step(f0-f,dx.norm());
|
|
if(!budget) //incomplete sweep, no convergence test
|
|
{
|
|
if(verbose) std::cout <<name<<": maximum number of function evaluations reached\n";
|
|
st.f=f;
|
|
break;
|
|
}
|
|
}
|
|
|
|
return st.finish(achieved);
|
|
}
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> SGD_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
SGD_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("SGD cannot search for saddle points");
|
|
SGD_stepper<T> stepper(hp);
|
|
return firstorder_minimize(hp.my_nesterov ? "SGD-Nesterov" : "SGD",stepper,o,x,hp,achieved);
|
|
}
|
|
|
|
|
|
//elementwise functions for Adam
|
|
template<typename T>
|
|
static T elementmax(const T &a, const T &b) {return a>b ? a : b;}
|
|
|
|
//m/(sqrt(v)/sqrtc2 + eps) with the bias correction sqrtc2 = sqrt(1-beta2^k)
|
|
template<typename T>
|
|
class adamratio {
|
|
public:
|
|
T sqrtc2, eps;
|
|
adamratio(const T s, const T e) : sqrtc2(s), eps(e) {};
|
|
T operator()(const T &m, const T &v) const {return m/(std::sqrt(v)/sqrtc2+eps);}
|
|
};
|
|
|
|
|
|
//Adam, AdamW, AMSGrad
|
|
template<typename T>
|
|
class Adam_stepper : public firstorder_step<T> {
|
|
public:
|
|
const Adam_hyperparameters<T> &hp;
|
|
NRVec<T> m; //first moment estimate
|
|
NRVec<T> v; //second moment estimate
|
|
NRVec<T> vmax; //maximum of the second moments for AMSGrad
|
|
T beta1k, beta2k; //beta1^k, beta2^k
|
|
Adam_stepper(const Adam_hyperparameters<T> &h) : hp(h), beta1k(1), beta2k(1) {};
|
|
NRVec<T> step(const NRVec<T> &g, const int k)
|
|
{
|
|
NRVec<T> g2(g);
|
|
g2 *= g;
|
|
if(k==1) {m = g*(1-hp.my_beta1); v = g2*(1-hp.my_beta2);}
|
|
else
|
|
{
|
|
m *= hp.my_beta1; m.axpy(1-hp.my_beta1,g);
|
|
v *= hp.my_beta2; v.axpy(1-hp.my_beta2,g2);
|
|
}
|
|
beta1k *= hp.my_beta1;
|
|
beta2k *= hp.my_beta2;
|
|
if(hp.my_amsgrad) {if(k==1) vmax=v; else vmax.call_on_me(elementmax<T>,v);}
|
|
NRVec<T> d = m*(1/(1-beta1k));
|
|
d.call_on_me(adamratio<T>(std::sqrt(1-beta2k),hp.my_eps), hp.my_amsgrad ? vmax : v);
|
|
return d;
|
|
}
|
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};
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template<typename T>
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optresult<T> Adam_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
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{
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Adam_hyperparameters<T> hp;
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if(par) hp.setfrom(*par);
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if(hp.my_saddle_order) laerror("Adam cannot search for saddle points");
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Adam_stepper<T> stepper(hp);
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return firstorder_minimize(hp.my_amsgrad ? "AMSGrad" : (hp.my_weight_decay!=0 ? "AdamW" : "Adam"),stepper,o,x,hp,achieved);
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}
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//SPSA gradient estimate (g_hat)_i = (f(x+c_k Delta) - f(x-c_k Delta))/(2 c_k Delta_i), Delta_i = +-1, averaged over resamplings
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template<typename T>
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class SPSA_estimator : public gradient_estimator<T> {
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public:
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const SPSA_hyperparameters<T> &hp;
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SPSA_estimator(const SPSA_hyperparameters<T> &h) : hp(h) {};
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T operator()(optevaluator<T> &e, const NRVec<T> &x, NRVec<T> &g, const int k);
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int cost() const {return 2*hp.my_resamplings;}
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bool exact() const {return false;}
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T calibrate(optevaluator<T> &e, const NRVec<T> &x, const firstorder_hyperparameters<T> &h);
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};
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//returns the average of f(x+c_k Delta) and f(x-c_k Delta) as an estimate of f(x)
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template<typename T>
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T SPSA_estimator<T>::operator()(optevaluator<T> &e, const NRVec<T> &x, NRVec<T> &g, const int k)
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{
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const int n=x.size();
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const T ck=hp.my_c/std::pow((T)(k+1),hp.my_gamma);
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g.clear();
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T fsum=0;
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for(int r=0; r<hp.my_resamplings; ++r)
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{
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NRVec<T> delta(n); //fresh vector, element access is safe
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for(int i=0; i<n; ++i) delta[i]= randdouble()<0.5 ? -1 : 1;
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NRVec<T> xp(x), xm(x);
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xp.axpy(ck,delta);
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xm.axpy(-ck,delta);
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T fp=e(xp);
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T fm=e(xm);
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g.axpy((fp-fm)/(2*ck),delta); //1/Delta_i = Delta_i
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fsum+=fp+fm;
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}
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g /= (T)hp.my_resamplings;
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return fsum/(2*hp.my_resamplings);
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}
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|
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//learning rate such that the first step lr*|g_hat| has on average the target length
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|
template<typename T>
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|
T SPSA_estimator<T>::calibrate(optevaluator<T> &e, const NRVec<T> &x, const firstorder_hyperparameters<T> &h)
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|
{
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|
if(hp.my_calibrate<=0) return hp.my_lr;
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|
T gsum=0;
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|
int ncal=0;
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|
for(; ncal<hp.my_calibrate && e.remaining()>=2*cost(); ++ncal) //leave the budget for the first estimate
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|
{
|
|
NRVec<T> g(x.size());
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|
(*this)(e,x,g,0);
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|
gsum+=g.norm();
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|
}
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|
if(ncal==0 || gsum==0) return hp.my_lr;
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|
return hp.my_target_step*ncal/gsum;
|
|
}
|
|
|
|
|
|
//plain gradient step for SPSA
|
|
template<typename T>
|
|
class plain_stepper : public firstorder_step<T> {
|
|
public:
|
|
NRVec<T> step(const NRVec<T> &g, const int k) {return g;}
|
|
};
|
|
|
|
|
|
template<typename T>
|
|
optresult<T> SPSA_Optimizer<T>::minimize(const OptProblem<T> *o, NRVec<T> &x, const hyperparameters<T> *par, hyperparameters<T> *achieved)
|
|
{
|
|
SPSA_hyperparameters<T> hp;
|
|
if(par) hp.setfrom(*par);
|
|
if(hp.my_saddle_order) laerror("SPSA cannot search for saddle points");
|
|
if(hp.my_resamplings<1) laerror("SPSA needs resamplings >= 1");
|
|
plain_stepper<T> stepper;
|
|
SPSA_estimator<T> est(hp);
|
|
return firstorder_minimize("SPSA",stepper,o,x,hp,achieved,&est);
|
|
}
|
|
|
|
|
|
|
|
//force instantiation
|
|
template double linesearch_wolfe(optevaluator<double> &e, NRVec<double> &x, double &f, NRVec<double> &g, const NRVec<double> &p, const double alpha1, const linesearch_hyperparameters<double> &lp);
|
|
template class optstate<double>;
|
|
template class testproblem<double>;
|
|
template class testproblem_vqe<double>;
|
|
template class testproblem_vqe_constrained<double>;
|
|
template class ParameterShift<double>;
|
|
template class FiniteDifference<double>;
|
|
template class BFGS_Optimizer<double>;
|
|
template class LBFGS_Optimizer<double>;
|
|
template class GDIIS_Optimizer<double>;
|
|
template class CG_Optimizer<double>;
|
|
template class NelderMead_Optimizer<double>;
|
|
template class Powell_Optimizer<double>;
|
|
template class Rotosolve_Optimizer<double>;
|
|
template class Cobyla_Optimizer<double>;
|
|
template class SGD_Optimizer<double>;
|
|
template class QNG_Optimizer<double>;
|
|
template class Adam_Optimizer<double>;
|
|
template class SPSA_Optimizer<double>;
|
|
template double linemin_brent(optevaluator<double> &e, NRVec<double> &x, double &f, const NRVec<double> &u, const double t1, const double tol, const double abstol, const int maxiter, const int verbose);
|
|
|
|
}//namespace
|
|
|