Bayesian Optimization

Goal

Efficiently find the optimal input parameters (e.g., alloy composition, simulation hyperparameters, process conditions) that minimize or maximize one or more expensive black-box objectives (e.g., formation energy, bandgap, elastic modulus) using Bayesian Optimization (BO). BO builds a probabilistic surrogate model (Gaussian Process) over the objective landscape and uses an acquisition function to intelligently select the next most informative experiments, minimizing the number of expensive evaluations required.

• Single-objective: Expected Improvement (EI) maximized via multi-start L-BFGS-B.
• Multi-objective: ParEGO — random Chebyshev scalarization with independent GPs, one weight vector per batch element, naturally steering candidates toward different Pareto-front regions.

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Instructions

Step 1: Define the Search Space

Create a search_space.yaml in the research directory. Use the template at resources/search_space_template.yaml as a starting point:

# research_dir/search_space.yaml

parameters:
  # Continuous range parameter
  - name: x_Fe
    type: range
    bounds: [0.0, 1.0]
    value_type: float

  # Integer range parameter
  - name: supercell_size
    type: range
    bounds: [2, 6]
    value_type: int

objectives:
  # Single-objective: minimize formation energy
  - name: formation_energy_eV_atom
    minimize: true

  # Multi-objective: additionally maximize bandgap (uncomment to enable)
  # - name: bandgap_eV
  #   minimize: false

Guidelines:

• Use type: range for continuous or integer parameters with known bounds. Only range parameters are passed to the GP surrogate.
• choice and fixed types are recorded in output CSVs but not optimized. Run separate campaigns per discrete choice.
• Choose bounds informed by domain knowledge; avoid unnecessarily wide ranges.

Step 2: Initialize with Quasi-Random Sobol Samples

Generate an initial space-filling design using Sobol sequences. Use a power-of-2 batch_size (4, 8, 16, …) for optimal Sobol balance:

# Env: base-agent
python .agents/skills/ml-bayesian-optimization/scripts/suggest_candidates.py \
    --config research_dir/search_space.yaml \
    --batch_size 8 \
    --output research_dir/candidates_round_0.csv \
    --output_dir research_dir/

With no --results provided (or fewer than 2 × N_range_params evaluated points), the script automatically uses Sobol initialization. The output CSV lists candidate parameter values only — it does not evaluate the objective.

Step 3: Evaluate Candidates Using MCP Tools

For each row in candidates_round_0.csv, call the appropriate MCP tool to evaluate the objective and record results. The script plays no role in this step.

Objective type	Recommended MCP tool
Energy / formation energy	mcp_mace_relax_structure or mcp_matgl_relax_structure
Bandgap	mcp_matgl_predict_bandgap
Arbitrary property	mcp_mace_predict_structure
DFT reference	mcp_atomate2_run_atomate2_vasp_calculation

Save all results to evaluated.csv — one row per candidate, parameter columns plus objective column(s):

x_Fe,formation_energy_eV_atom
0.12,-0.087
0.45,-0.143
0.73,-0.091

Step 4: BO Loop — Suggest Next Candidates

With evaluated results, fit the GP surrogate and suggest the next batch:

# Env: base-agent
python .agents/skills/ml-bayesian-optimization/scripts/suggest_candidates.py \
    --config research_dir/search_space.yaml \
    --results research_dir/evaluated.csv \
    --batch_size 4 \
    --output research_dir/candidates_round_1.csv \
    --output_dir research_dir/

The script will:

1. Normalize all inputs to [0, 1] per dimension.
2. Fit a GP with Matern-5/2 kernel (scikit-learn, normalize_y=True).
3. Single-objective: maximize EI via multi-start L-BFGS-B with diversity enforcement.
4. Multi-objective: fit one GP per batch element with a random Chebyshev weight vector (ParEGO).
5. Write the next batch_size candidates to the output CSV.
6. Append round metadata to campaign_state.json in --output_dir.

Repeat Steps 3–4 until the budget is exhausted or convergence is observed.

Step 5: Convergence Check and Analysis

# Env: base-agent
python .agents/skills/ml-bayesian-optimization/scripts/plot_bo_results.py \
    --results research_dir/evaluated.csv \
    --config research_dir/search_space.yaml \
    --output_dir research_dir/

This generates:

• convergence_curve.png — best observed value vs. evaluation number (flat line = converged)
• parameter_importance.png — scatter of each parameter vs. objective across all evaluations
• pareto_front.png — (multi-objective only) non-dominated Pareto front
• gp_model_1d.png — (single range parameter only) GP mean ± 2σ and EI landscape
• gp_model_2d.png — (exactly two range parameters) GP mean, uncertainty, and EI as contour maps

Convergence criteria (manual inspection):

• Single-objective: best value improves by < 1% over the last 5 rounds.
• Multi-objective: hypervolume of the Pareto front changes by < 2% over the last 5 rounds.

Visual Inspection: Use mcp_base_visualize_structure to inspect the best-found structure, and inspect all generated plots.

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Examples

Synthetic 2D Test (Branin Function)

Validate the BO workflow on a known 2D benchmark with three global minima at $f^* = 0.3979$:

# Env: base-agent
python .agents/skills/ml-bayesian-optimization/scripts/suggest_candidates.py \
    --config .agents/skills/ml-bayesian-optimization/examples/branin-function/search_space.yaml \
    --batch_size 8 \
    --output /tmp/bo_test/candidates_round_0.csv \
    --output_dir /tmp/bo_test/

See branin-function example for the full walkthrough including all 8 rounds and the GP model visualization.

Li-Ag Phase Stability (1D Composition Search)

Locate the most stable Li-Ag binary composition using Materials Project formation energies as the oracle:

See li-ag-phases example for the full walkthrough.

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Constraints

• Environment: base-agent. Dependencies: scikit-learn, scipy, numpy, pandas, pyyaml. No BoTorch or ax-platform required.
• GP Scaling: Training is O(n³). Practical upper limit is ~500 evaluated points before training time becomes noticeable.
• Batch Size: Use a power-of-2 for initialization (batch_size = 4, 8, 16, …). For BO rounds, batch_size ≤ 8 is recommended; larger is fine when evaluations are embarrassingly parallel.
• Minimum Data: The GP needs at least 2 × N_range_params evaluated points to fit reliably. The Sobol initialization (Step 2) should provide at least this many.
• Parameter Bounds: Bounds in search_space.yaml must be physically meaningful. The GP has no information about the objective outside the specified domain.
• Noise Handling: For deterministic objectives (analytic functions, noiseless ML models) set --noise_std 0. For stochastic objectives (MD-derived properties, noisy experiments) set --noise_std <estimated_std>.

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References

• Frazier, P.I., "A Tutorial on Bayesian Optimization", arXiv, 2018. arXiv:1807.02811
• Knowles, J., "ParEGO: A Hybrid Algorithm with On-Line Landscape Approximation for Expensive Multiobjective Optimization Problems", IEEE Transactions on Evolutionary Computation, 2006. DOI:10.1109/TEVC.2005.851274
• Balachandran, P.V. et al., "Adaptive strategies for materials design using uncertainties", Scientific Reports, 2016. DOI:10.1038/srep19660
• Lookman, T. et al., "Active learning in materials science with emphasis on adaptive sampling using uncertainties for targeted design", npj Computational Materials, 2019. DOI:10.1038/s41524-019-0153-8

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Author: Bowen Deng Contact: GitHub @bowen-bd