The Modular CMA-ES: A Framework for Modern Evolution Strategies

• Because variants are often studied in isolation, component interactions are difficult to understand and practitioners must navigate specialized implementations and interfaces.The effect of a sampling strategy may depend on matrix adaptation or restart choices.Source: §1 · pp. 1–2
• ModCMA provides fourteen interchangeable modules in one implementation, spanning sampling, selection, step-size adaptation, matrix adaptation, and restarting.The framework integrates many CMA-ES variants within a component-based architecture.Source: §1 · pp. 2–3
• Cholesky- and matrix-based updates can match or surpass the standard covariance update while substantially reducing runtime across BBOB benchmark comparisons.The comparison quantifies computational trade-offs alongside optimization performance.Source: §1 · p. 3
• Per-function SMAC tuning on BBOB produced specialized variants, while SHAP analysis was used to examine how module choices contribute to tuned performance.The configurations were compared with established algorithms from the COCO archives.Source: §1 · p. 3
• The authors provide ModCMA as an open-source package under an MIT license and make code, data, and additional figures available for reproducibility.The code and data are also available in a Zenodo repository.Source: §1 · p. 3

2.1 The CMA-ES Algorithm

• CMA-ES samples a population from a multivariate normal distribution and recombines selected candidates to update its mean, step size, and covariance.Step-size adaptation controls mutation scale, while covariance adaptation reshapes sampling toward successful search directions.Source: §2.1 · p. 3
• The algorithm’s five functional components—restarting, sampling, selection and recombination, step-size adaptation, and matrix adaptation—are conceptually separable despite sharing state.Each component can admit alternative implementations.Source: §2.1 · p. 4

2.2 From CMA-ES to Modular CMA-ES

• ModCMA exposes interchangeable options for CMA-ES components; a configuration specifies module options together with numerical parameter settings.The framework follows the modular evolutionary algorithms paradigm and extends an earlier modular CMA-ES implementation.Source: §2.2 · p. 5

2.3 A Modular Structure

• ModCMA implements fourteen modules, and some options depend on others: Active Update has no effect without matrix adaptation, while Repelling is inactive without restarts.The framework distinguishes modular variants by combinations of module options when numerical parameters remain at prescribed defaults.Source: §2.3 · pp. 5–6

3 Sampling

• Sampling draws standard-normal mutation vectors, transforms them by A, then scales them by the mean and global step size.ModCMA implements five modules for the sampling component.Source: §3 · p. 6

3.1 Base Sampler

• Halton and Sobol low-discrepancy sequences can replace the standard pseudorandom source of uniform values used to construct mutation vectors.The text motivates these sequences by noting that pseudorandom numbers can be unevenly distributed, especially for smaller samples.Source: §3.1 · p. 6

3.2 Sample Transformation

• Six functional distributions can replace the Gaussian transformation of uniform samples, while the maximum-entropy Gaussian is the default.Source: §3.2 · p. 6

3.3 Mirrored Sampling

• Mirrored sampling adds sign-reversed mutation vectors, and pairwise selection uses only the better member of each mirrored pair in the parameter update.With weighted recombination and CSA, mirrored sampling can bias toward smaller step sizes and potentially premature convergence.Source: §3.3 · pp. 6–7

3.4 Orthogonal Sampling

• Orthogonal sampling uses Gram–Schmidt orthogonalization to distribute offspring more evenly, with prior BBOB comparisons reporting better local convergence and performance than mirrored sampling.Source: §3.4 · p. 7

3.5 Threshold Convergence

• Threshold Convergence enforces a generation-dependent minimum distance from the population mean by pushing sampled points outward when they fall within the threshold.The method aims to improve exploration, particularly on complex multimodal problems.Source: §3.5 · p. 7

3.6 Boundary Correction

• ModCMA implements seven boundary-correction options for out-of-domain samples, and repaired points require corresponding intermediate values to be recalculated for parameter updates.The listed strategies include resampling, reflection, uniform replacement, one-tailed normal replacement, saturation, and toroidal correction.Source: §3.6 · pp. 7–8

4 Selection and Recombination

• CMA-ES generates λ offspring and generally uses the µ best individuals for weighted intermediate recombination.Source: §4 · p. 8

4.1 Elitism

• Elitism uses (µ + λ) selection to consider parents alongside offspring and preserve the best solution found, while non-elitist selection is generally preferred for stronger exploration.Solutions from a previous generation must be transformed appropriately for parameter updates under elitism.Source: §4.1 · p. 8

4.2 Recombination Weights

• Recombination weights may decrease logarithmically by rank, be equal, or decrease exponentially; active updates also require negative weights for unselected individuals.Source: §4.2 · p. 8

4.3 Sequential Selection

• Sequential selection can stop evaluating a generation once at least µ solutions are evaluated and an improvement is observed, reducing evaluations but requiring sequential rather than parallel evaluation.Source: §4.3 · pp. 8–9

5.1 Step-Size Adaptation

• CSA can produce too-small step sizes and premature convergence with noisy objectives or ineffective coordinates; ModCMA implements eight alternatives.The cited discussion also identifies large population sizes as a setting where CSA struggles.Source: §5.1 · p. 9
• TPA avoids a predefined optimality criterion but requires two extra evaluations and bases step-size adaptation only on those evaluations.It resembles self-adaptation at the population level.Source: §5.1 · p. 9
• SR smooths improvements in the generation’s best individual toward a target success rate of 2/11, while MSR tracks median improvement toward 1/2.PSR instead uses rank differences between current and previous populations to govern step-size updates.Source: §5.1 · pp. 9–10
• xNES adapts step size from weighted normalized mutation vectors, while m-XNES uses movement of the search-distribution mean and lp-xNES and SA sample individual step sizes.The sampled step sizes in lp-xNES and SA come from log-normal distributions centered on the current global step size.Source: §5.1 · p. 10

5.2 Matrix Adaptation

• Full covariance adaptation requires an O(d^3) spectral decomposition, motivating ModCMA’s six alternative matrix-adaptation strategies.When CSA does not require C^-1/2, ModCMA uses Cholesky decomposition, which remains cubic but has a smaller constant factor in practice.Source: §5.2 · p. 11
• Disabling covariance adaptation leaves C = I and can remain effective on separable, well-conditioned objectives; diagonal adaptation is linear-time and can help on ill-conditioned problems.Diagonal adaptation can also outperform CMA-ES on strictly separable functions.Source: §5.2 · p. 11
• MA-ES learns a transformation matrix without explicit covariance maintenance or expensive decomposition and retains comparable performance; Cholesky CMA-ES uses triangular factors, while CMSA removes evolution paths.The Cholesky approach has theoretical arguments that its use of A^-1 for the step-size path does not impair performance; CMSA uses a less stable covariance estimator and increased selective pressure by default.Source: §5.2 · pp. 11–12

5.3 Active Update

• Active-CMA-ES assigns negative weights to the worst-performing candidates, actively reducing variance in their directions; ModCMA makes this update optional.For xNES, the active update sums over all λ individuals rather than the default µ.Source: §5.3 · p. 13

6 Restarting

• Restart criteria can check internal parameters such as σ and C or stagnation in observed objective values, and users can disable criteria, adjust tolerances, or specify custom criteria.A triggered criterion can stop or restart CMA-ES; subsequent behavior is controlled by the restart-strategy modules.Source: §6 · p. 13

6.1 Restart Strategy

• IPOP starts with the default population size and doubles λ at each restart, retaining the small initial population for simple problems and increasing it for complex multimodal landscapes.The population size is capped at 2^9λ0.Source: §6.1 · p. 14
• BIPOP alternates an increasing-population regime with a smaller, randomly parameterized regime, selecting at each restart the regime with fewer total evaluations.The small regime randomizes its step size; the paper describes BIPOP as a top-performing BBOB algorithm.Source: §6.1 · p. 14

6.2 Repelling

• Repelling Restarts maintains an archive of previously visited basins and repels new candidates from those regions, combining with the described restart schemes.This contrasts with native restart strategies, which typically ignore information from previous runs.Source: §6.2 · p. 14

7.1 (Mixed)-Integer Optimization

• For integer variables, ModCMA lower-bounds the effective step size and rounds sampled values to the nearest integer, maintaining variation along integer dimensions.The implementation uses only the lower-bounding component of the cited approach.Source: §7.1 · pp. 14–15

7.2 The (1+1)-CMA-ES

• Setting λ = 1 activates (1+1)-CMA-ES, which compares one offspring with its parent and updates distribution parameters only after offspring success.This mode fixes or disables several modules, including elitist selection, equal recombination weights, and success-rule step-size adaptation.Source: §7.2 · p. 15

7.3 ConfigSpace

• The ConfigSpace interface exposes categorical module choices and selected numerical parameters, enabling tuning with HPO tools such as SMAC.Population-size parameters are log-scaled integers constrained by µ0 ≤ λ0; other included parameters are bounded continuous values centered on defaults.Source: §7.3 · p. 15

8 A closer look at Matrix Adaptation

• The experiments use official CMA-ES from Pycma v4.2.0 as the baseline; for ModCMA v1.1.0, only matrix adaptation is modified, with other options at their defaults.The objective is a d-dimensional ellipsoid (BBOB f2), with 15 runs for each d in {2, 3, 5, 10, 20, 50, 100, 200, 500, 1 000}. Runtime was measured using Python 3.10.12 on a desktop running Windows subsystem for Linux 6.6.87.1 with a 12th Gen Intel i5-12400F CPU.Source: §8 · p. 16, §8.1 · p. 16

8.1 Running Time

• Per matrix update, standard CMA is least efficient; across tested dimensions, Cholesky CMA significantly outperforms Pycma in runtime.For d > 500, Pycma is notably faster than ModCMA’s Covariance update, while the two fastest methods, None and Separable, are indistinguishable in Figure 1.Source: Figure 1 · p. 17, §8.1 · p. 17

8.2 Benchmarking

• ModCMA’s Covariance results align well with Pycma across most benchmarks, with minor differences on several multimodal functions.The authors attribute variability on multimodal functions as likely due to randomness in sample-based optimization.Source: §8.2 · p. 18
• The Separable strategy performs well on separable f2 but poorly on rotated, non-separable f10; no matrix adaptation performs well on f1 and f6.No matrix adaptation also seems to perform comparably on f17 and f21, where matrix adaptation may be less important.Source: §8.2 · p. 18
• Cholesky and Matrix adaptation show no significant performance drop across most BBOB benchmarks and find f13 optimizers for d > 10 where Covariance and Pycma do not.Both are described as viable alternatives to the more computationally demanding Covariance method.Source: §8.2 · p. 18, Figure 2 · p. 19
• Without restarts and with default population size, many benchmarks are unsolvable in moderately high dimensions within the configured budget.On f24, almost no runs reach 10^-9, so meaningful conclusions cannot be drawn; comparisons are informative mainly where configurations reliably reach the target.Source: §8.2 · p. 20

9.1 Tuning Setup

• SMAC tunes each BBOB function independently, and the five best candidates per function are validated on 50 previously unused instances.The reported tuned candidate is the one with the lowest validation-set ERT.Source: §9.1 · pp. 20–21

9.2 Benchmark Results

• SMAC configurations substantially reduce ERT on most functions, while the default often fails to reach the target on f3, f4, and f15–f24.For f15–f23, tuning yields finite ERT where the default has very low success; the configurator fails to identify a reliable solver only for f4 and f24.Source: §9.2 · p. 21
• Tuned ModCMA matches or improves on the bbob2009 reference on half the functions, with 2.5–5× gains on f15–f20.The comparison uses ERT across 24 BBOB functions in dimension 5; Figure 3 also shows default and tuned configurations against the per-function reference solver.Source: Figure 3 · p. 22, §9.2 · p. 22

9.3 Interpreting the Configuration Space

• Global SHAP results associate stronger performance influence with Step-Size Adaptation than with repelling restarts, but aggregated effects depend on function-specific benefits and interactions.The analysis uses median option contributions across evaluated configurations; the authors caution that interactions can make options beneficial in isolation appear detrimental on average.Source: §9.3 · p. 23, Figure 4 · p. 23
• Matrix-adaptation SHAP contributions vary by function: disabling adaptation is detrimental on conditioned f2 and f10, while its choice has limited influence on highly multimodal f4 and f24.Cholesky-based and standard covariance adaptation show the most consistently positive contributions across functions.Source: §9.3 · pp. 23–24, Figure 5 · p. 24

10 Conclusion

• ModCMA exposes interchangeable algorithmic components in one implementation, enabling systematic combination, benchmarking, and configuration of CMA-ES variants.The framework covers sampling, selection, step-size adaptation, matrix adaptation, and restarting.Source: §10 · p. 24
• Cholesky- and Matrix-based adaptation can reduce runtime while retaining competitive performance on many benchmark problems.The matrix-adaptation study identifies trade-offs between computational cost and optimization performance.Source: §10 · p. 24
• Per-function SMAC tuning substantially improves on default ModCMA and outperforms the bbob2009 reference on half of the functions; SHAP analysis reveals global and problem-specific option effects.The conclusion identifies future work in extending modules, supporting mixed-integer and constrained optimization, and landscape-aware selection.Source: §10 · pp. 24–25

A Expected running time

• Figure 6 plots ERT for default and tuned ModCMA alongside individual algorithms from the bbob2009 portfolio across 24 functions in dimension 5.Lower ERT indicates better performance.Source: Figure 6 · p. 34
• The bold entry for each function marks the lowest ERT among the compared algorithms.Source: Figure 6 · p. 34
• Large or infinite ERT values indicate that an algorithm failed to reach the target within the evaluation budget.Source: Figure 6 · p. 34

B Per Module SHAP contributions

• Figures 7–10 map Active Update, Elitism, Mirrored Sampling, and Orthogonal Sampling options across the 24 BBOB functions.Rows are module options; columns are functions, and cell medians indicate tendency to improve or degrade performance.Source: Figure 7 · p. 35, Figure 8 · p. 35, Figure 9 · p. 35
• Figures 11–14 show per-function SHAP contributions for Repelling, Restart Strategy, Sample Transformation, and Base Sampler options.Negative cell values indicate a tendency to improve performance; positive values indicate a tendency to degrade it.Source: Figure 11 · p. 36, Figure 12 · p. 36, Figure 13 · p. 36
• Figures 15–18 cover Sequential Selection, Step-Size Adaptation, Threshold Convergence, and Recombination Weights across the function set.Each cell reports the median SHAP value over evaluated configurations for that option and function.Source: Figure 15 · p. 37, Figure 16 · p. 37, Figure 17 · p. 38