curvopt 0.1.0

Curvature-aware optimization with exact Hessians via hcderiv

CAO — Curvature-Aware Optimization

CAO provides practical second-order optimization using exact Hessians computed via hcderiv.

Make trust-region and Newton-style optimization as easy to use as Adam or SGD.

No finite differences. No approximations. No step-size tuning.

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Install

pip install curvopt                  # NumPy backend
pip install "cao[jax]"           # + JAX/XLA backend (recommended)

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Quick start

from cao import TrustRegionOptimizer
import numpy as np

def f(X):
    return (X[1] - X[0]**2)**2 * 100.0 + (X[0]*0.0 + 1.0 - X[0])**2

opt = TrustRegionOptimizer(f, backend="jax-xla")
result = opt.minimize([-1.2, 1.0])

print(result)
# OptimizeResult(f=1.5e-15, ‖g‖∞=3.4e-08, nit=23, converged)

print(result.x)
# [1. 1.]

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How it works

At each iteration CAO calls hcderiv.grad_and_hessian(f, x, backend=...) to compute the exact gradient and full Hessian in a single forward pass. It then solves the trust-region subproblem:

min  g·p + ½ p·H·p   s.t.  ‖p‖ ≤ Δ

using the Cauchy point as a safe fallback and the regularised Newton step when it lies inside the trust region. The radius Δ is adapted each iteration based on the ratio of actual to predicted reduction.

Why exact Hessians?

Quasi-Newton methods (L-BFGS) and diagonal approximations miss off-diagonal curvature. hcderiv's XLA backend computes the full d×d Hessian in one forward pass with NumPy-class speed, making exact second-order steps practical for the first time at this scale.

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Architecture

hcderiv (exact curvature engine)
    ↓  grad_and_hessian(f, x, backend="jax-xla")
cao (trust-region solver)
    ↓  TrustRegionOptimizer.minimize(x0)
x*

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Performance

Rosenbrock from (-1.2, 1.0) — convergence to f < 1e-14:

Method	Iterations	Hessian evals
CAO trust-region (exact H)	23	23
scipy L-BFGS-B	~120	0 (approx)
scipy Newton-CG	~35	35 (FD)

CAO converges in fewer iterations because it uses the exact curvature at every step.

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API

from cao import TrustRegionOptimizer, TrustRegionConfig, OptimizeResult

# Configure
config = TrustRegionConfig(
    max_iters=200,
    initial_radius=1.0,
    tol_grad=1e-6,
    verbose=True,       # print iteration log
)

# Run
opt = TrustRegionOptimizer(f, config=config, backend="jax-xla")
result = opt.minimize(x0)

# Result fields
result.x           # final iterate
result.f           # function value
result.grad_norm   # ‖∇f‖∞ at x*
result.nit         # iterations
result.nfev        # Hessian evaluations
result.converged   # bool
result.f_history   # f at each accepted step

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Ecosystem

hcderiv          →  exact Hessians (NumPy + JAX-XLA)
    ↓
cao              →  curvature-aware optimizer
    ↓
constitutional-os + governed-research-lab-v2  →  curvature-governed agents

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Roadmap

• v0.2.0 — cubic regularization (ARC)
• v0.3.0 — PyTorch backend
• v0.4.0 — curvature-aware learning rate schedules for ML models

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Citation

A JOSS paper will accompany CAO after the 6-month public availability window. Until then, please cite the hcderiv engine:

@misc{byte2026d,
  author = {Byte, Zetta},
  title  = {hcderiv v0.4.0 — JAX-XLA backend for exact one-pass Hessians},
  year   = {2026},
  doi    = {10.5281/zenodo.19433812},
  url    = {https://doi.org/10.5281/zenodo.19433812}
}