Atomizer/docs/guides/CMA-ES_EXPLAINED.md

CMA-ES Explained for Engineers

CMA-ES = Covariance Matrix Adaptation Evolution Strategy

A derivative-free optimization algorithm ideal for:

• Local refinement around known good solutions
• 4-10 dimensional problems
• Smooth, continuous objective functions
• Problems where gradient information is unavailable (like FEA)

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The Core Idea

Imagine searching for the lowest point in a hilly landscape while blindfolded:

1. Throw darts around your current best guess
2. Observe which darts land lower (better objective)
3. Learn the shape of the valley from those results
4. Adjust future throws to follow the valley's direction

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Key Components

┌─────────────────────────────────────────────────────────────┐
│                      CMA-ES Components                       │
├─────────────────────────────────────────────────────────────┤
│                                                             │
│  1. MEAN (μ) - Current best guess location                  │
│     • Moves toward better solutions each generation         │
│                                                             │
│  2. STEP SIZE (σ) - How far to throw darts                  │
│     • Adapts: shrinks when close, grows when exploring      │
│     • sigma0=0.3 means 30% of parameter range initially     │
│                                                             │
│  3. COVARIANCE MATRIX (C) - Shape of the search cloud       │
│     • Learns parameter correlations                         │
│     • Stretches search along promising directions           │
│                                                             │
└─────────────────────────────────────────────────────────────┘

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Visual: How the Search Evolves

Generation 1 (Round search):        Generation 10 (Learned shape):

    x    x                              x
      x    x                          x   x
    x  ●  x     ──────►            x    ●    x
      x    x                          x   x
    x    x                              x

  ● = mean (center)                Ellipse aligned with
  x = samples                      the valley direction

CMA-ES learns that certain parameter combinations work well together and stretches its search cloud in that direction.

---

The Algorithm (Simplified)

def cma_es_generation():
    # 1. SAMPLE: Generate λ candidates around the mean
    for i in range(population_size):
        candidates[i] = mean + sigma * sample_from_gaussian(covariance=C)

    # 2. EVALUATE: Run FEA for each candidate
    for candidate in candidates:
        fitness[candidate] = run_simulation(candidate)

    # 3. SELECT: Keep the best μ candidates
    selected = top_k(candidates, by=fitness, k=mu)

    # 4. UPDATE MEAN: Move toward the best solutions
    new_mean = weighted_average(selected)

    # 5. UPDATE COVARIANCE: Learn parameter correlations
    C = update_covariance(C, selected, mean, new_mean)

    # 6. UPDATE STEP SIZE: Adapt exploration range
    sigma = adapt_step_size(sigma, evolution_path)

---

The Covariance Matrix Magic

Consider 4 design variables:

Covariance Matrix C (4x4):
                    var1    var2    var3    var4
var1               [ 1.0     0.3    -0.5     0.1 ]
var2               [ 0.3     1.0     0.2    -0.2 ]
var3               [-0.5     0.2     1.0     0.4 ]
var4               [ 0.1    -0.2     0.4     1.0 ]

Reading the matrix:

• Diagonal (1.0): Variance in each parameter
• Off-diagonal: Correlations between parameters
• Positive (0.3): When var1 increases, var2 should increase
• Negative (-0.5): When var1 increases, var3 should decrease

CMA-ES learns these correlations automatically from simulation results!

---

CMA-ES vs TPE

Property	TPE	CMA-ES
Best for	Global exploration	Local refinement
Starting point	Random	Known baseline
Correlation learning	None (independent)	Automatic
Step size	Fixed ranges	Adaptive
Dimensionality	Good for high-D	Best for 4-10D
Sample efficiency	Good	Excellent (locally)

---

Optuna Configuration

from optuna.samplers import CmaEsSampler

# Baseline values (starting point)
x0 = {
    'whiffle_min': 62.75,
    'whiffle_outer_to_vertical': 75.89,
    'whiffle_triangle_closeness': 65.65,
    'blank_backface_angle': 4.43
}

sampler = CmaEsSampler(
    x0=x0,           # Center of initial distribution
    sigma0=0.3,      # Initial step size (30% of range)
    seed=42,         # Reproducibility
    restart_strategy='ipop'  # Increase population on restart
)

study = optuna.create_study(sampler=sampler, direction="minimize")

# CRITICAL: Enqueue baseline as trial 0!
# x0 only sets the CENTER, it doesn't evaluate the baseline
study.enqueue_trial(x0)

study.optimize(objective, n_trials=200)

---

Common Pitfalls

1. Not Evaluating the Baseline

Problem: CMA-ES samples AROUND x0, but doesn't evaluate x0 itself.

Solution: Always enqueue the baseline:

if len(study.trials) == 0:
    study.enqueue_trial(x0)

2. sigma0 Too Large or Too Small

sigma0	Effect
Too large (>0.5)	Explores too far, misses local optimum
Too small (<0.1)	Gets stuck, slow convergence
Recommended (0.2-0.3)	Good balance for refinement

3. Wrong Problem Type

CMA-ES struggles with:

• Discrete/categorical variables
• Very high dimensions (>20)
• Multi-modal landscapes (use TPE first)
• Noisy objectives (add regularization)

---

When to Use CMA-ES in Atomizer

Scenario	Use CMA-ES?
First exploration of design space	No, use TPE
Refining around known good design	Yes
4-10 continuous variables	Yes
>15 variables	No, use TPE or NSGA-II
Need to learn variable correlations	Yes
Multi-objective optimization	No, use NSGA-II

---

References

• Hansen, N. (2016). The CMA Evolution Strategy: A Tutorial
• Optuna CmaEsSampler: https://optuna.readthedocs.io/en/stable/reference/samplers/generated/optuna.samplers.CmaEsSampler.html
• cmaes Python package: https://github.com/CyberAgentAILab/cmaes

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Created: 2025-12-19 Atomizer Framework