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Optimizer: update the dense BFGS inverse Hessian in O(n²) instead of O(n³) #3443

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Optimizer: update the dense BFGS inverse Hessian in O(n²) instead of O(n³) #3443

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8c8426f
Update the dense BFGS inverse Hessian in O(n^2)
jachymb Oct 17, 2026
88df557
Keep only the lower triangle of the dense BFGS inverse Hessian
avehtari Oct 18, 2026
fd3f0b2
Merge branch 'develop' into feature/bfgs-update-rank-two
jachymb Oct 25, 2026
1ae9420
Merge branch 'develop' into feature/bfgs-update-rank-two
jachymb Oct 27, 2026
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22 changes: 15 additions & 7 deletions src/stan/optimization/bfgs_update.hpp
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Original file line number Diff line number Diff line change
Expand Up @@ -25,21 +25,28 @@ class BFGSUpdate_HInv {
inline Scalar update(const VectorT &yk, const VectorT &sk,
bool reset = false) {
Scalar rhok, skyk, B0fact;
HessianT Hupd;

skyk = yk.dot(sk);
rhok = 1.0 / skyk;

Hupd.noalias()
= HessianT::Identity(yk.size(), yk.size()) - rhok * sk * yk.transpose();
if (reset) {
B0fact = yk.squaredNorm() / skyk;
_Hk.noalias() = ((1.0 / B0fact) * Hupd) * Hupd.transpose();
_Hk.setIdentity(yk.size(), yk.size());
_Hk /= B0fact;
} else {
B0fact = 1.0;
_Hk = Hupd * _Hk * Hupd.transpose();
}
_Hk.noalias() += rhok * sk * sk.transpose();
// H <- (I - rho s y') H (I - rho y s') + rho s s'
// = H + s w' + w s', u = H y, w = (c / 2) s - rho u,
// c = rho^2 y'u + rho
// Only the lower triangle of _Hk is kept up to date.
_u.noalias() = _Hk.template selfadjointView<Eigen::Lower>() * yk;
const Scalar c = rhok * rhok * yk.dot(_u) + rhok;
_w = (0.5 * c) * sk - rhok * _u;
// rankUpdate copies plain vector arguments; Maps avoid the allocations
_Hk.template selfadjointView<Eigen::Lower>().rankUpdate(
Eigen::Map<const VectorT>(sk.data(), sk.size()),
Eigen::Map<const VectorT>(_w.data(), _w.size()));
return B0fact;
}

Expand All @@ -52,11 +59,12 @@ class BFGSUpdate_HInv {
* @param[in] gk Gradient direction.
**/
inline void search_direction(VectorT &pk, const VectorT &gk) const {
pk.noalias() = -(_Hk * gk);
pk.noalias() = _Hk.template selfadjointView<Eigen::Lower>() * (-gk);
}

private:
HessianT _Hk;
VectorT _u, _w;
};
} // namespace optimization
} // namespace stan
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