amcpso — Adaptive Mirror-Consensus Particle Swarm Optimization

amcpso is a gradient-assisted particle swarm optimizer that evolves particles in the dual space of a strongly convex mirror potential. This buys two things classical PSO doesn't have:

• Feasibility by construction. With the entropic mirror, every iterate is automatically a valid point on the probability simplex (x_k >= 0, sum(x) == 1) — no clip-and-renormalize projection distorting the search.
• Entropy-regulated annealing. A swarm-dispersion signal r_t in [0, 1] closes a feedback loop that cools exploration (noise, consensus pull) automatically as the swarm converges, instead of using a fixed schedule.

It combines five forces per iteration — inertia, natural-gradient descent, mirrored cognitive/social attraction, graph-Laplacian consensus over a time-varying neighborhood topology, and annealed Gaussian noise.

Installation

pip install amcpso

Requires Python >= 3.8 and NumPy >= 1.20.

When to use it (and when not to)

Use amcpso when your problem is constrained, geometry-sensitive, and/or multimodal:

• Simplex-constrained problems — recovering or optimizing over a probability distribution, mixture weights, portfolio allocations, or any compositional/proportion data (h_type='entropic'). This is where the mirror machinery pays off most: feasibility is structural, not statistical, and the algorithm is significantly better than projection-based PSO in the reference benchmarks.
• Rugged, multimodal landscapes in ordinary R^d (h_type='euclidean'), such as the Rastrigin function. The entropy-regulated schedule and local (rather than global-best) topology help delay premature convergence to a poor local minimum, and it is competitive-to-better than classical constriction PSO in the reference benchmarks.
• You have (or can cheaply compute, e.g. via autodiff or finite differences) a gradient of the objective — amcpso is gradient-assisted, not a purely black-box method.

Don't expect a win when:

• Your objective is smooth and unimodal (e.g. a plain quadratic bowl) — classical PSO's noiseless contraction is typically competitive or better here, consistent with the No-Free-Lunch theorem.
• You have no usable gradient and computing one is expensive or impossible — consider a derivative-free method instead.

Quick start

import numpy as np
from amcpso import amc_pso

def sphere(x):
    return float(np.sum(x**2))

def sphere_grad(x):
    return 2 * x

x_best, f_best = amc_pso(sphere, sphere_grad, d=5, seed=0)
print(x_best, f_best)

The function: amc_pso(...)

from amcpso import amc_pso

x_best, f_best = amc_pso(
    f, grad_f, d,
    N=50, T=500, h_type='euclidean', init_range=(-5, 5),
    beta=0.7, eta0=0.05, c10=1.5, c20=1.5, lambda0=0.3,
    T0=0.5, alpha=5.0, gamma=1.0, bandwidth=2.0, epsilon=1e-10,
    kappa=1e-3, p=0.5, G_max=10.0, v_max=None,
    natural='diag', naive=False, entropy_feedback=True,
    return_history=False, seed=None,
)

Required arguments

Key optional arguments

Full parameter-by-parameter documentation is available in the function's docstring: help(amc_pso).

Returns

• x_best (numpy.ndarray, shape (d,)) — best position found.
• f_best (float) — objective value at x_best.
• history (dict, only if return_history=True) — per-iteration traces.

Example: multimodal optimization (Rastrigin, R^d)

This is the best-use-case scenario for the Euclidean mirror: a rugged, multimodal landscape.

import numpy as np
from amcpso import amc_pso
from amcpso.benchmarks import rastrigin, rastrigin_grad

d = 10
x_best, f_best = amc_pso(
    rastrigin, rastrigin_grad, d=d,
    h_type='euclidean', init_range=(-5.12, 5.12),
    N=50, T=500, seed=7,
)
print(f"best objective: {f_best:.4f}")

Example: constrained optimization on the probability simplex

This is the best use case for amcpso: recovering a probability vector x* on the simplex from noisy linear measurements b = A @ x*, where every iterate must satisfy x >= 0 and sum(x) == 1.

import numpy as np
from amcpso import amc_pso

rng = np.random.default_rng(12345)
d = 20
A = rng.standard_normal((40, d))
x_star = rng.dirichlet(np.ones(d))   # ground truth on the simplex
b = A @ x_star

def loss(x):
    return float(np.sum((A @ x - b) ** 2))

def loss_grad(x):
    return 2.0 * A.T @ (A @ x - b)

x_best, f_best = amc_pso(loss, loss_grad, d=d, h_type='entropic', seed=0)

print("sum(x_best) =", x_best.sum())      # 1.0, always -- feasible by construction
print("min(x_best) =", x_best.min())      # >= 0, always
print("loss        =", f_best)

Unlike classical PSO with clip-and-renormalize projection, amc_pso with h_type='entropic' never produces an infeasible iterate at any point during the search — the softmax inverse mirror map guarantees it structurally.

Diagnosing convergence with return_history

x_best, f_best, hist = amc_pso(
    rastrigin, rastrigin_grad, d=10,
    h_type='euclidean', init_range=(-5.12, 5.12),
    seed=7, return_history=True,
)

# hist['best']: best-so-far objective per iteration
# hist['r']:    dispersion ratio r_t in [0, 1]
# hist['T']:    temperature T_t
# hist['lam']:  consensus gain lambda_t
# hist['eta']:  step size eta_t

Algorithm summary

Each particle keeps a primal position x, a dual variable y = grad_h(x), and a dual velocity v. Every iteration:

1. Rebuilds a time-varying neighborhood topology from a thresholded Gaussian kernel on pairwise distances.
2. Measures swarm dispersion S_t (half log-determinant of the swarm covariance) and derives a dispersion ratio r_t = min(1, exp((S_t - S_0) / d)), which drives the consensus gain and temperature schedules.
3. Updates the dual velocity as the sum of inertia, natural-gradient descent, cognitive attraction, social attraction, Laplacian consensus, and annealed noise.
4. Maps back to the primal space with the inverse mirror map (x = grad_h_star(y)), which is the identity for the Euclidean mirror and a softmax for the entropic mirror.

See the function docstring (help(amc_pso)) and the accompanying manuscript for the full derivation.

License

MIT — see LICENSE.