GLE Colored-Noise Langevin¶

GLE-Langevin replaces the white-noise Langevin move with a generalized Langevin equation thermostat. An optimal-sampling drift matrix fitted once from the characteristic frequency flattens the noise spectrum. The same fitted matrix accelerates every gradient-capable driver.

Reference

The BAB propagator, stationary reseed per epoch, matrix_exp + ldl_sqrt construction of the drift, and the reduction to the canonical form appear in src/methods/gle_langevin.rs and the corresponding eindir GleThermostat / optimal_sampling_drift code. The “one drift serves all gradient drivers” claim is a direct consequence of the algebra (the GLE object occupies exactly the Move slot).

Prerequisites¶

Gradient of the objective required. Bounds.

pip install anneal

Step 1: Define obj + grad¶

import numpy as np
from anneal import gle_langevin

def rastrigin(x):
    return 10.0 * len(x) + np.sum(x*x - 10.0 * np.cos(2.0 * np.pi * x))

def grad_rastrigin(x):
    return 2.0 * x + 20.0 * np.pi * np.sin(2.0 * np.pi * x)

low = np.full(5, -5.0)
high = np.full(5, 5.0)

Step 2: Call `gle_langevin` (the exposed driver)¶

res = gle_langevin(rastrigin, grad_rastrigin, low, high,
                   max_fevals=4000, seed=11,
                   omega0=0.2, dt=0.2, n_epochs=40)
print("GLE best pos:", res["best_pos"])
print("GLE best val:", res["best_val"])
print("GLE evals:", res["n_evals"])

Expected on this separable ill-conditioned problem: the colored-noise version reaches lower values in the same budget than plain white-noise Langevin would, because the single fitted drift matrix equalizes efficiency across a decade of frequencies.

Step 3: The thermostat inside (what the algebra buys)¶

eindir builds the optimal_sampling_drift matrix A for the chosen omega0.
The GLE move kernel (B-A-B splitting + exact matrix exponential; the propagator from the GLE literature) occupies the Move slot.
The same A works for any inner sampler that supplies a gradient (HMC variants, plain Langevin, custom methods).
No per-preset GLE code exists; the factoring guarantees it.

Stationary reseed per epoch keeps the auxiliary momenta consistent with the invariant measure even when the outer temperature schedule changes.

Why this works¶

White-noise Langevin critically damps only one frequency. Real objectives have spectra spanning orders of magnitude (the classic “ill-conditioned” case). The GLE drift matrix is the exact continuous-time solution that makes the integrated autocorrelation time flat from omega0 to ~100*=omega0=. Because Move is the only slot that knows the proposal shape, swapping the white-noise proposal for the GLE proposal gives gradient drivers the same thermostat path.

See gle-mechanics (T4) for the matrix construction and the canonical-form derivation.