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CMA-ES Optimizer

How MatCraft uses Covariance Matrix Adaptation Evolution Strategy for materials optimization.

Apr 1, 20268 min read
cma-es
optimizer
evolution-strategy

CMA-ES Optimizer

CMA-ES (Covariance Matrix Adaptation Evolution Strategy) is the default optimization algorithm in MatCraft. It is a derivative-free, stochastic optimization method particularly well-suited for non-convex, multi-modal fitness landscapes common in materials science.

Why CMA-ES?

Materials optimization problems have several properties that make CMA-ES an excellent choice:

  • No gradient required: Evaluation functions are often black-box physics simulations or experimental measurements with no analytical gradient.
  • Non-convex landscapes: Material property surfaces frequently have multiple local optima.
  • Moderate dimensionality: Most material design spaces have 5—50 parameters, which is the sweet spot for CMA-ES.
  • Noisy evaluations: CMA-ES is robust to evaluation noise from simulations or experiments.

Algorithm Overview

CMA-ES maintains a multivariate normal distribution over the parameter space and iteratively adapts it:

  1. Sample a population of candidate solutions from the current distribution N(m, sigma^2 * C), where m is the mean, sigma is the step size, and C is the covariance matrix.
  2. Evaluate each candidate using the surrogate model (or directly via the evaluation function).
  3. Rank candidates by fitness (objective value or acquisition function score).
  4. Update the distribution parameters:
  • The mean m moves toward the best candidates.
  • The covariance matrix C adapts to the local fitness landscape shape.
  • The step size sigma adjusts based on the evolution path.
  1. Repeat until budget is exhausted or convergence is detected.

Integration with Surrogate Models

In MatCraft, CMA-ES does not directly evaluate the expensive physics model. Instead, it optimizes the acquisition function over the surrogate model:

CMA-ES proposes candidates
    -> Acquisition function (EI) scores candidates using surrogate predictions
    -> Best candidates are selected for expensive evaluation
    -> Surrogate is retrained with new data
    -> CMA-ES restarts with updated surrogate

This two-level approach is far more sample-efficient than running CMA-ES directly on the evaluation function.

Configuration

CMA-ES parameters are set in the optimizer section of the MDL file:

yaml
optimizer:
  method: cma-es
  budget: 300
  batch_size: 15
  sigma0: 0.3
  population_size: 50

| Parameter | Default | Description | |—————-|————-|——————-| | sigma0 | 0.3 | Initial step size (relative to normalized bounds). Larger values encourage exploration; smaller values focus near initial solutions. | | population_size | auto | Population size per CMA-ES generation. Default is 4 + floor(3 * ln(n)) where n is the number of parameters. |

Handling Different Parameter Types

CMA-ES operates natively on continuous variables. MatCraft handles the other types transparently:

  • Continuous parameters: Normalized to [0, 1] and sampled directly.
  • Integer parameters: Sampled as continuous, then rounded to the nearest integer.
  • Categorical parameters: One-hot encoded. CMA-ES samples in the continuous relaxation, and the highest-scoring category is selected via argmax.

Multi-Objective Extension

For multi-objective optimization, MatCraft uses CMA-ES to optimize the acquisition function, which itself accounts for multi-objective trade-offs via Expected Hypervolume Improvement (EHVI). The CMA-ES population is scored by EHVI rather than a single fitness value.

Restart Strategy

CMA-ES can converge prematurely to a local optimum. MatCraft implements an automatic restart strategy:

  • When the step size sigma drops below a threshold (1e-8), CMA-ES restarts with an increased initial sigma.
  • Each restart doubles the population size (IPOP-CMA-ES strategy).
  • The restart preserves all previously evaluated data for surrogate training.

Performance Characteristics

| Design Space Size | Expected Performance | |—————————|——————————-| | 2—5 parameters | Excellent. Converges in 50—100 evaluations. | | 5—20 parameters | Very good. Converges in 100—300 evaluations. | | 20—50 parameters | Good. May require 300—1000 evaluations. | | 50—100 parameters | Moderate. Consider dimensionality reduction. | | 100+ parameters | Not recommended. Use alternative methods. |

Programmatic Access

python
from materia.optimize.cmaes import CMAESOptimizer

optimizer = CMAESOptimizer(
    dim=5,
    sigma0=0.3,
    population_size=20,
    seed=42,
)

# Propose candidates
candidates = optimizer.ask(n=10)

# Report fitness values
optimizer.tell(candidates, fitness_values)

See Also