462 lines
18 KiB
C++
462 lines
18 KiB
C++
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/*
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* Copyright Nick Thompson, 2018
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* Use, modification and distribution are subject to the
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* Boost Software License, Version 1.0. (See accompanying file
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* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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*/
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#ifndef BOOST_MATH_QUADRATURE_NAIVE_MONTE_CARLO_HPP
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#define BOOST_MATH_QUADRATURE_NAIVE_MONTE_CARLO_HPP
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#include <sstream>
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#include <algorithm>
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#include <vector>
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#include <atomic>
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#include <memory>
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#include <functional>
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#include <future>
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#include <thread>
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#include <initializer_list>
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#include <utility>
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#include <random>
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#include <chrono>
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#include <map>
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#include <type_traits>
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#include <boost/math/policies/error_handling.hpp>
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namespace boost { namespace math { namespace quadrature {
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namespace detail {
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enum class limit_classification {FINITE,
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LOWER_BOUND_INFINITE,
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UPPER_BOUND_INFINITE,
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DOUBLE_INFINITE};
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}
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template<class Real, class F, class RandomNumberGenerator = std::mt19937_64, class Policy = boost::math::policies::policy<>,
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typename std::enable_if<std::is_trivially_copyable<Real>::value, bool>::type = true>
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class naive_monte_carlo
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{
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public:
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naive_monte_carlo(const F& integrand,
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std::vector<std::pair<Real, Real>> const & bounds,
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Real error_goal,
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bool singular = true,
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uint64_t threads = std::thread::hardware_concurrency(),
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uint64_t seed = 0) noexcept : m_num_threads{threads}, m_seed{seed}
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{
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using std::numeric_limits;
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using std::sqrt;
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uint64_t n = bounds.size();
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m_lbs.resize(n);
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m_dxs.resize(n);
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m_limit_types.resize(n);
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m_volume = 1;
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static const char* function = "boost::math::quadrature::naive_monte_carlo<%1%>";
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for (uint64_t i = 0; i < n; ++i)
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{
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if (bounds[i].second <= bounds[i].first)
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{
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boost::math::policies::raise_domain_error(function, "The upper bound is <= the lower bound.\n", bounds[i].second, Policy());
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return;
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}
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if (bounds[i].first == -numeric_limits<Real>::infinity())
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{
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if (bounds[i].second == numeric_limits<Real>::infinity())
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{
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m_limit_types[i] = detail::limit_classification::DOUBLE_INFINITE;
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}
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else
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{
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m_limit_types[i] = detail::limit_classification::LOWER_BOUND_INFINITE;
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// Ok ok this is bad to use the second bound as the lower limit and then reflect.
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m_lbs[i] = bounds[i].second;
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m_dxs[i] = numeric_limits<Real>::quiet_NaN();
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}
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}
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else if (bounds[i].second == numeric_limits<Real>::infinity())
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{
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m_limit_types[i] = detail::limit_classification::UPPER_BOUND_INFINITE;
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if (singular)
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{
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// I've found that it's easier to sample on a closed set and perturb the boundary
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// than to try to sample very close to the boundary.
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m_lbs[i] = std::nextafter(bounds[i].first, (std::numeric_limits<Real>::max)());
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}
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else
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{
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m_lbs[i] = bounds[i].first;
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}
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m_dxs[i] = numeric_limits<Real>::quiet_NaN();
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}
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else
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{
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m_limit_types[i] = detail::limit_classification::FINITE;
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if (singular)
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{
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if (bounds[i].first == 0)
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{
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m_lbs[i] = std::numeric_limits<Real>::epsilon();
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}
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else
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{
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m_lbs[i] = std::nextafter(bounds[i].first, (std::numeric_limits<Real>::max)());
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}
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m_dxs[i] = std::nextafter(bounds[i].second, std::numeric_limits<Real>::lowest()) - m_lbs[i];
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}
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else
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{
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m_lbs[i] = bounds[i].first;
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m_dxs[i] = bounds[i].second - bounds[i].first;
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}
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m_volume *= m_dxs[i];
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}
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}
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m_integrand = [this, &integrand](std::vector<Real> & x)->Real
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{
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Real coeff = m_volume;
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for (uint64_t i = 0; i < x.size(); ++i)
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{
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// Variable transformation are listed at:
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// https://en.wikipedia.org/wiki/Numerical_integration
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// However, we've made some changes to these so that we can evaluate on a compact domain.
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if (m_limit_types[i] == detail::limit_classification::FINITE)
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{
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x[i] = m_lbs[i] + x[i]*m_dxs[i];
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}
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else if (m_limit_types[i] == detail::limit_classification::UPPER_BOUND_INFINITE)
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{
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Real t = x[i];
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Real z = 1/(1 + numeric_limits<Real>::epsilon() - t);
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coeff *= (z*z)*(1 + numeric_limits<Real>::epsilon());
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x[i] = m_lbs[i] + t*z;
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}
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else if (m_limit_types[i] == detail::limit_classification::LOWER_BOUND_INFINITE)
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{
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Real t = x[i];
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Real z = 1/(t+sqrt((numeric_limits<Real>::min)()));
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coeff *= (z*z);
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x[i] = m_lbs[i] + (t-1)*z;
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}
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else
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{
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Real t1 = 1/(1+numeric_limits<Real>::epsilon() - x[i]);
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Real t2 = 1/(x[i]+numeric_limits<Real>::epsilon());
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x[i] = (2*x[i]-1)*t1*t2/4;
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coeff *= (t1*t1+t2*t2)/4;
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}
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}
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return coeff*integrand(x);
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};
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// If we don't do a single function call in the constructor,
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// we can't do a restart.
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std::vector<Real> x(m_lbs.size());
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// If the seed is zero, that tells us to choose a random seed for the user:
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if (seed == 0)
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{
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std::random_device rd;
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seed = rd();
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}
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RandomNumberGenerator gen(seed);
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Real inv_denom = 1/static_cast<Real>(((gen.max)()-(gen.min)()));
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m_num_threads = (std::max)(m_num_threads, static_cast<uint64_t>(1));
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m_thread_calls.reset(new std::atomic<uint64_t>[threads]);
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m_thread_Ss.reset(new std::atomic<Real>[threads]);
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m_thread_averages.reset(new std::atomic<Real>[threads]);
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Real avg = 0;
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for (uint64_t i = 0; i < m_num_threads; ++i)
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{
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for (uint64_t j = 0; j < m_lbs.size(); ++j)
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{
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x[j] = (gen()-(gen.min)())*inv_denom;
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}
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Real y = m_integrand(x);
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m_thread_averages[i] = y; // relaxed store
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m_thread_calls[i] = 1;
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m_thread_Ss[i] = 0;
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avg += y;
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}
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avg /= m_num_threads;
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m_avg = avg; // relaxed store
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m_error_goal = error_goal; // relaxed store
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m_start = std::chrono::system_clock::now();
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m_done = false; // relaxed store
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m_total_calls = m_num_threads; // relaxed store
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m_variance = (numeric_limits<Real>::max)();
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}
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std::future<Real> integrate()
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{
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// Set done to false in case we wish to restart:
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m_done.store(false); // relaxed store, no worker threads yet
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m_start = std::chrono::system_clock::now();
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return std::async(std::launch::async,
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&naive_monte_carlo::m_integrate, this);
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}
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void cancel()
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{
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// If seed = 0 (meaning have the routine pick the seed), this leaves the seed the same.
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// If seed != 0, then the seed is changed, so a restart doesn't do the exact same thing.
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m_seed = m_seed*m_seed;
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m_done = true; // relaxed store, worker threads will get the message eventually
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// Make sure the error goal is infinite, because otherwise we'll loop when we do the final error goal check:
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m_error_goal = (std::numeric_limits<Real>::max)();
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}
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Real variance() const
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{
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return m_variance.load();
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}
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Real current_error_estimate() const
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{
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using std::sqrt;
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//
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// There is a bug here: m_variance and m_total_calls get updated asynchronously
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// and may be out of synch when we compute the error estimate, not sure if it matters though...
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//
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return sqrt(m_variance.load()/m_total_calls.load());
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}
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std::chrono::duration<Real> estimated_time_to_completion() const
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{
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auto now = std::chrono::system_clock::now();
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std::chrono::duration<Real> elapsed_seconds = now - m_start;
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Real r = this->current_error_estimate()/m_error_goal.load(); // relaxed load
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if (r*r <= 1) {
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return 0*elapsed_seconds;
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}
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return (r*r - 1)*elapsed_seconds;
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}
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void update_target_error(Real new_target_error)
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{
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m_error_goal = new_target_error; // relaxed store
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}
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Real progress() const
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{
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Real r = m_error_goal.load()/this->current_error_estimate(); // relaxed load
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if (r*r >= 1)
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{
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return 1;
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}
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return r*r;
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}
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Real current_estimate() const
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{
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return m_avg.load();
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}
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uint64_t calls() const
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{
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return m_total_calls.load(); // relaxed load
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}
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private:
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Real m_integrate()
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{
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uint64_t seed;
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// If the user tells us to pick a seed, pick a seed:
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if (m_seed == 0)
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{
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std::random_device rd;
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seed = rd();
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}
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else // use the seed we are given:
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{
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seed = m_seed;
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}
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RandomNumberGenerator gen(seed);
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int max_repeat_tries = 5;
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do{
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if (max_repeat_tries < 5)
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{
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m_done = false;
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#ifdef BOOST_NAIVE_MONTE_CARLO_DEBUG_FAILURES
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std::cout << "Failed to achieve required tolerance first time through..\n";
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std::cout << " variance = " << m_variance << std::endl;
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std::cout << " average = " << m_avg << std::endl;
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std::cout << " total calls = " << m_total_calls << std::endl;
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for (std::size_t i = 0; i < m_num_threads; ++i)
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std::cout << " thread_calls[" << i << "] = " << m_thread_calls[i] << std::endl;
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for (std::size_t i = 0; i < m_num_threads; ++i)
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std::cout << " thread_averages[" << i << "] = " << m_thread_averages[i] << std::endl;
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for (std::size_t i = 0; i < m_num_threads; ++i)
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std::cout << " thread_Ss[" << i << "] = " << m_thread_Ss[i] << std::endl;
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#endif
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}
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std::vector<std::thread> threads(m_num_threads);
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for (uint64_t i = 0; i < threads.size(); ++i)
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{
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threads[i] = std::thread(&naive_monte_carlo::m_thread_monte, this, i, gen());
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}
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do {
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std::this_thread::sleep_for(std::chrono::milliseconds(100));
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uint64_t total_calls = 0;
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for (uint64_t i = 0; i < m_num_threads; ++i)
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{
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uint64_t t_calls = m_thread_calls[i].load(std::memory_order_consume);
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total_calls += t_calls;
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}
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Real variance = 0;
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Real avg = 0;
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for (uint64_t i = 0; i < m_num_threads; ++i)
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{
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uint64_t t_calls = m_thread_calls[i].load(std::memory_order_consume);
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// Will this overflow? Not hard to remove . . .
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avg += m_thread_averages[i].load(std::memory_order_relaxed)*(static_cast<Real>(t_calls) / static_cast<Real>(total_calls));
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variance += m_thread_Ss[i].load(std::memory_order_relaxed);
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}
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m_avg.store(avg, std::memory_order_release);
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m_variance.store(variance / (total_calls - 1), std::memory_order_release);
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m_total_calls = total_calls; // relaxed store, it's just for user feedback
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// Allow cancellation:
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if (m_done) // relaxed load
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{
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break;
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}
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} while (m_total_calls < 2048 || this->current_error_estimate() > m_error_goal.load(std::memory_order_consume));
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// Error bound met; signal the threads:
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m_done = true; // relaxed store, threads will get the message in the end
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std::for_each(threads.begin(), threads.end(),
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std::mem_fn(&std::thread::join));
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if (m_exception)
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{
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std::rethrow_exception(m_exception);
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}
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// Incorporate their work into the final estimate:
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uint64_t total_calls = 0;
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for (uint64_t i = 0; i < m_num_threads; ++i)
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{
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uint64_t t_calls = m_thread_calls[i].load(std::memory_order_consume);
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total_calls += t_calls;
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}
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Real variance = 0;
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Real avg = 0;
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for (uint64_t i = 0; i < m_num_threads; ++i)
|
||
|
|
{
|
||
|
|
uint64_t t_calls = m_thread_calls[i].load(std::memory_order_consume);
|
||
|
|
// Averages weighted by the number of calls the thread made:
|
||
|
|
avg += m_thread_averages[i].load(std::memory_order_relaxed)*(static_cast<Real>(t_calls) / static_cast<Real>(total_calls));
|
||
|
|
variance += m_thread_Ss[i].load(std::memory_order_relaxed);
|
||
|
|
}
|
||
|
|
m_avg.store(avg, std::memory_order_release);
|
||
|
|
m_variance.store(variance / (total_calls - 1), std::memory_order_release);
|
||
|
|
m_total_calls = total_calls; // relaxed store, this is just user feedback
|
||
|
|
|
||
|
|
// Sometimes, the master will observe the variance at a very "good" (or bad?) moment,
|
||
|
|
// Then the threads proceed to find the variance is much greater by the time they hear the message to stop.
|
||
|
|
// This *WOULD* make sure that the final error estimate is within the error bounds.
|
||
|
|
}
|
||
|
|
while ((--max_repeat_tries >= 0) && (this->current_error_estimate() > m_error_goal));
|
||
|
|
|
||
|
|
return m_avg.load(std::memory_order_consume);
|
||
|
|
}
|
||
|
|
|
||
|
|
void m_thread_monte(uint64_t thread_index, uint64_t seed)
|
||
|
|
{
|
||
|
|
using std::numeric_limits;
|
||
|
|
try
|
||
|
|
{
|
||
|
|
std::vector<Real> x(m_lbs.size());
|
||
|
|
RandomNumberGenerator gen(seed);
|
||
|
|
Real inv_denom = static_cast<Real>(1) / static_cast<Real>(( (gen.max)() - (gen.min)() ));
|
||
|
|
Real M1 = m_thread_averages[thread_index].load(std::memory_order_consume);
|
||
|
|
Real S = m_thread_Ss[thread_index].load(std::memory_order_consume);
|
||
|
|
// Kahan summation is required or the value of the integrand will go on a random walk during long computations.
|
||
|
|
// See the implementation discussion.
|
||
|
|
// The idea is that the unstabilized additions have error sigma(f)/sqrt(N) + epsilon*N, which diverges faster than it converges!
|
||
|
|
// Kahan summation turns this to sigma(f)/sqrt(N) + epsilon^2*N, and the random walk occurs on a timescale of 10^14 years (on current hardware)
|
||
|
|
Real compensator = 0;
|
||
|
|
uint64_t k = m_thread_calls[thread_index].load(std::memory_order_consume);
|
||
|
|
while (!m_done) // relaxed load
|
||
|
|
{
|
||
|
|
int j = 0;
|
||
|
|
// If we don't have a certain number of calls before an update, we can easily terminate prematurely
|
||
|
|
// because the variance estimate is way too low. This magic number is a reasonable compromise, as 1/sqrt(2048) = 0.02,
|
||
|
|
// so it should recover 2 digits if the integrand isn't poorly behaved, and if it is, it should discover that before premature termination.
|
||
|
|
// Of course if the user has 64 threads, then this number is probably excessive.
|
||
|
|
int magic_calls_before_update = 2048;
|
||
|
|
while (j++ < magic_calls_before_update)
|
||
|
|
{
|
||
|
|
for (uint64_t i = 0; i < m_lbs.size(); ++i)
|
||
|
|
{
|
||
|
|
x[i] = (gen() - (gen.min)())*inv_denom;
|
||
|
|
}
|
||
|
|
Real f = m_integrand(x);
|
||
|
|
using std::isfinite;
|
||
|
|
if (!isfinite(f))
|
||
|
|
{
|
||
|
|
// The call to m_integrand transform x, so this error message states the correct node.
|
||
|
|
std::stringstream os;
|
||
|
|
os << "Your integrand was evaluated at {";
|
||
|
|
for (uint64_t i = 0; i < x.size() -1; ++i)
|
||
|
|
{
|
||
|
|
os << x[i] << ", ";
|
||
|
|
}
|
||
|
|
os << x[x.size() -1] << "}, and returned " << f << std::endl;
|
||
|
|
static const char* function = "boost::math::quadrature::naive_monte_carlo<%1%>";
|
||
|
|
boost::math::policies::raise_domain_error(function, os.str().c_str(), /*this is a dummy arg to make it compile*/ 7.2, Policy());
|
||
|
|
}
|
||
|
|
++k;
|
||
|
|
Real term = (f - M1)/k;
|
||
|
|
Real y1 = term - compensator;
|
||
|
|
Real M2 = M1 + y1;
|
||
|
|
compensator = (M2 - M1) - y1;
|
||
|
|
S += (f - M1)*(f - M2);
|
||
|
|
M1 = M2;
|
||
|
|
}
|
||
|
|
m_thread_averages[thread_index].store(M1, std::memory_order_release);
|
||
|
|
m_thread_Ss[thread_index].store(S, std::memory_order_release);
|
||
|
|
m_thread_calls[thread_index].store(k, std::memory_order_release);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
catch (...)
|
||
|
|
{
|
||
|
|
// Signal the other threads that the computation is ruined:
|
||
|
|
m_done = true; // relaxed store
|
||
|
|
std::lock_guard<std::mutex> lock(m_exception_mutex); // Scoped lock to prevent race writing to m_exception
|
||
|
|
m_exception = std::current_exception();
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
std::function<Real(std::vector<Real> &)> m_integrand;
|
||
|
|
uint64_t m_num_threads;
|
||
|
|
std::atomic<uint64_t> m_seed;
|
||
|
|
std::atomic<Real> m_error_goal;
|
||
|
|
std::atomic<bool> m_done;
|
||
|
|
std::vector<Real> m_lbs;
|
||
|
|
std::vector<Real> m_dxs;
|
||
|
|
std::vector<detail::limit_classification> m_limit_types;
|
||
|
|
Real m_volume;
|
||
|
|
std::atomic<uint64_t> m_total_calls;
|
||
|
|
// I wanted these to be vectors rather than maps,
|
||
|
|
// but you can't resize a vector of atomics.
|
||
|
|
std::unique_ptr<std::atomic<uint64_t>[]> m_thread_calls;
|
||
|
|
std::atomic<Real> m_variance;
|
||
|
|
std::unique_ptr<std::atomic<Real>[]> m_thread_Ss;
|
||
|
|
std::atomic<Real> m_avg;
|
||
|
|
std::unique_ptr<std::atomic<Real>[]> m_thread_averages;
|
||
|
|
std::chrono::time_point<std::chrono::system_clock> m_start;
|
||
|
|
std::exception_ptr m_exception;
|
||
|
|
std::mutex m_exception_mutex;
|
||
|
|
};
|
||
|
|
|
||
|
|
}}}
|
||
|
|
#endif
|