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#include "pcg-cpp/include/pcg_random.hpp"
#include "randutils/randutils.hpp"
#include "eigen/Eigen/Dense"
using Rng = randutils::random_generator<pcg32>;
using Real = double;
using Vector = Eigen::Matrix<Real, Eigen::Dynamic, 1>;
using Matrix = Eigen::Matrix<Real, Eigen::Dynamic, Eigen::Dynamic>;
using Data = std::vector<std::tuple<Real, Real>>;
#ifndef FITS_ABS_BASIS
inline Real basisFunctions(unsigned N, unsigned i, Real x) {
return std::legendre(i, 2.0 * (x - 0.5));
}
#endif
#ifdef FITS_ABS_BASIS
inline Real basisFunctions(unsigned N, unsigned i, Real x) {
return abs(x - i / (N - 1.0));
}
#endif
Real value(const Vector& coeffs, Real x) {
Real v = 0;
for (unsigned j = 0; j < coeffs.size(); j++) {
v += coeffs[j] * basisFunctions(coeffs.size(), j, x);
}
return v;
}
Real cost(Data::const_iterator batchBegin, Data::const_iterator batchEnd, const Vector& coeffs) {
Real c = 0;
#pragma omp parallel for reduction(+:c)
for (Data::const_iterator it = batchBegin; it != batchEnd; it++) {
Real x = std::get<0>(*it);
Real y = std::get<1>(*it);
c += 0.5 * pow(value(coeffs, x) - y, 2);
}
return c;
}
Vector dCost(Data::const_iterator batchBegin, Data::const_iterator batchEnd, const Vector& coeffs) {
Vector dC = Vector::Zero(coeffs.size());
for (Data::const_iterator it = batchBegin; it != batchEnd; it++) {
Real x = std::get<0>(*it);
Real y = std::get<1>(*it);
Real Δc = value(coeffs, x) - y;
#pragma omp parallel for
for (unsigned j = 0; j < coeffs.size(); j++) {
dC[j] += Δc * basisFunctions(coeffs.size(), j, x);
}
}
return dC;
}
Vector underSolve(const Data& data, unsigned N) {
unsigned M = data.size();
Matrix A(M, N);
Vector b(M);
for (unsigned i = 0; i < M; i++) {
Real x = std::get<0>(data[i]);
Real y = std::get<1>(data[i]);
b[i] = y;
for (unsigned j = 0; j < N; j++) {
A(i, j) = basisFunctions(N, j, x);
}
}
return A.colPivHouseholderQr().solve(b);
}
Vector stochasticGradientDescent(const Data& data, const Vector& a₀, unsigned nBatches, long unsigned maxSteps, Real ε = 1e-12) {
Vector xₜ = a₀;
Real Hₜ;
Real α = 1.0;
Real m;
Vector ∇H;
long unsigned steps = 0;
unsigned batchSize = data.size() / nBatches;
Data::const_iterator batchBegin = data.begin();
Data::const_iterator batchEnd = data.begin() + batchSize;
Data::const_iterator effectiveEnd = data.begin() + batchSize * nBatches;
while (
Hₜ = cost(batchBegin, batchEnd, xₜ),
∇H = dCost(batchBegin, batchEnd, xₜ),
m = ∇H.squaredNorm(),
m > ε && steps < maxSteps
) {
Vector xₜ₊₁;
Real Hₜ₊₁;
while (
xₜ₊₁ = xₜ - α * ∇H, Hₜ₊₁ = cost(batchBegin, batchEnd, xₜ₊₁),
Hₜ₊₁ > Hₜ - 0.25 * α * m
) {
α /= 2;
}
xₜ = xₜ₊₁;
α *= 1.25;
steps++;
if (batchEnd == data.end()) {
batchBegin = data.begin();
batchEnd = data.begin() + batchSize;
} else if (batchEnd == effectiveEnd) {
batchBegin = effectiveEnd;
batchEnd = data.end();
} else {
batchBegin += batchSize;
batchEnd += batchSize;
}
}
return xₜ;
}
Data generateData(Real(*f)(Real), unsigned M, Real σ, Rng& r) {
Data data;
data.reserve(M);
for (unsigned i = 0; i < M; i++) {
Real x = ((Real)i) / (M - 1.0);
data.push_back({x, f(x) + r.variate<Real, std::normal_distribution>(0, σ)});
}
return data;
}
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