Overview
CMA-ES samples candidate solutions from a multivariate normal distribution and updates its mean and covariance to follow successful search directions.
SpecializedSource checked
CMA-ES
Optimizes continuous functions by adapting a Gaussian search distribution.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Optimization
- Difficulty
- Advanced
- Family
- Evolution Strategy
- Tier
- Specialized
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 2
- Code example
- Ready
- Visualization
- Ready
- Aliases
- Covariance Matrix Adaptation Evolution Strategy
Core Concepts
covariance adaptationsearch distributionevolution pathcontinuous optimization
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n^2)
Average
O(iterations * population * n^2)
Worst
O(iterations * population * n^2)
Space
O(n^2)
When to Use
- Use when tuning model parameters needs covariance adaptation.
- Use when the input matches the assumptions behind evolution Strategy.
When Not to Use
- Avoid when the objective is cheap and convex enough for a simpler exact solver.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Optimizes continuous functions by adapting a Gaussian search distribution
- Uses covariance adaptation and search distribution to guide the work.
- Produces best candidate.
Use Cases
- Tuning model parameters
- Searching design spaces
- Fitting models to noisy data
Inputs
- Objective function
- Initial candidate or population
- Stopping rule
Outputs
- Best candidate
- Objective value
- Search history
Key Formulas (1)
Sampling rule
xk = mean + σ · Normal(0, C)
x_k = mean + sigma * Normal(0, C)Draws candidates from the current covariance-shaped search distribution.
Pseudocode
- 1.Prepare the inputs and state needed for CMA-ES.
- 2.Apply the covariance adaptation step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return best candidate.
Canonical Example
- Input
- CMA-ES input with a small representative case
- Output
- Best candidate
- Explanation
- CMA-ES applies covariance adaptation to produce best candidate.
Implementation Notes
- Track both best-so-far and current candidate state.
- Keep the covariance adaptation step separate enough to test independently.
Common Pitfalls
- Stopping too early can make a poor local result look like convergence.
- Forgetting to check edge cases can make CMA-ES look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryPlot candidate points, objective values, and improvement steps across iterations.
Step 1 of 4: Prepare the inputs and state needed for CMA-ES.
optimization-mechanismNotes
CMA-ES is powerful for derivative-free continuous optimization but can become expensive as dimensionality grows.
Sources (1)
What each reference supports for this entry.
CMA-ES tutorial
PaperTutorial reference for covariance matrix adaptation evolution strategy.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageAlternative toDifferential Evolution
Both are derivative-free optimizers for continuous spaces.
Conceptual neighborGenetic Algorithm
CMA-ES is an evolutionary strategy rather than a crossover-focused genetic algorithm.
Related Algorithms
Tags
OptimizationEvolution StrategyPopulation BasedStochastic