AIMech AI

IPOP-CMA-ES

Algorithms AIMech:IPOPCMAES DRAFT
The original increasing-population restart CMA-ES algorithm, which independently restarts its weighted, rank-mu covariance-matrix-adaptation inner optimizer with doubled population size after local termination, using fresh initialization, inherited cumulative step-size adaptation and separate overall stopping conditions.
Identity Notes & Concept Scope This record describes the original 2005 named whole IPOP-CMA-ES algorithm with the cited 2004 positive-weight rank-mu and cumulative step-size base. The independent population-doubling restarts are additional to the maintained 2001 CMA-ES scope. The 2003 covariance component alone, IGO/xNES correspondences, local fixed-population restarts and later BIPOP are distinct. Fresh postmerge collection at a7303d92795cc371954c527372fce9117342d7b5 checked all 735 native and 442 pinned AIO record identities and bytes. Four searches included ignored and hidden files, yielding 1,529 occurrences grouped into 45 exact contents, all reconciled with earlier research; root read all 45 complete contents. No normalized ID, label, alias or scoped-synonym collision was found for the explicit candidate probes. This is bounded to the pinned inventory and inspected semantic neighborhoods, not an exhaustive absence claim about AI literature. Existing frontier owner 202 is resolved in place without adding a duplicate lead.

Claims & Grounded Evidence

6 Claims
Mechanism #weighted-gaussian-cma-inner-search
The original IPOP procedure uses the cited 2004 weighted, rank-mu CMA-ES as its inner optimizer. At each generation it draws lambda independent candidates from a Gaussian with mean m and covariance sigma squared C, evaluates the objective for minimization, and sets the new mean to a positive weighted combination of the best mu candidates, with weights summing to one. The covariance rule combines a rank-one evolution-path contribution with a rank-mu contribution from selected displacements. The method uses objective evaluations rather than objective derivatives; the Gaussian is a search distribution, not a predictive model of objective values.
Mechanism #inherited-cumulative-step-size-adaptation
The fixed inner routine adapts its scalar step size using a conjugate evolution path. With old-generation covariance C=B*D^2*B^T, it accumulates the normalized mean displacement after whitening by B*D^(-1)*B^T into p_sigma. It then uses sigma_new=sigma*exp((c_sigma/d_sigma)*(norm(p_sigma_new)/E_norm_standard_normal-1)), where the expectation is the length of an n-dimensional standard-normal vector. This scalar exponential is inherited cumulative step-size adaptation, distinct from the population-doubling restart rule. A positive initial sigma remains positive for finite exponents in exact real arithmetic; this does not guarantee objective improvement or floating-point stability.
Mechanism #independent-population-doubling-restarts
When an inner attempt meets a local stopping criterion, launch an independent restart with twice its population size. The original procedure starts with lambda_0=4+floor(3 ln n), so the population of an actually launched attempt indexed k from zero is lambda_k=2^k*lambda_0. Each attempt uses the default parameter formulas for its population: mu=floor(lambda/2), positive recombination weights and dependent effective-selection and learning-rate parameters are recomputed accordingly. Reusing default formulas does not mean freezing every derived numerical constant. The doubling occurs between attempts, not every generation, and no indefinitely continuing sequence is implied under a finite overall budget.
Mechanism #corrected-local-restart-triggers
The stated local termination categories are a stagnant or sufficiently small function-value range, small coordinate spread and evolution-path scale, finite-precision no-effect tests, or a covariance condition number above 10^14. For the no-effect-coordinate test, apply the explicit author correction: restart if adding 0.2 times the coordinate standard deviation leaves at least one coordinate of the mean unchanged. The uncorrected does-change wording is not the rule. These triggers indicate stagnation or numerical limits; they are not certificates that the global optimum has been reached.
Scope #fresh-initialization-and-overall-stopping
In the evaluated 2005 configuration, each restart samples a fresh initial mean uniformly in the prescribed box [A,B]^n and sets sigma_0=(B-A)/2; the cited base starts C at the identity and both evolution paths at zero. The benchmark specifies an overall allowance of n*10^4 function evaluations or termination when the objective error relative to the known optimum is below 10^-8. These overall conditions are separate from the local restart triggers. The box, initial scale, target, budget and boundary policy are configuration choices, not a requirement that an arbitrary black-box problem have a known optimum or this exact budget.
Scope #original-whole-method-and-local-comparator-boundary
The named original IPOP algorithm comprises its increasing-population restart policy and specified weighted, rank-mu, evolution-path and step-size-adapting CMA base. It is not every restart controller or an exact alias of all CMA-ES versions. The companion local-restart CMA-ES keeps the same small default population across independent attempts. Its evaluated initial scale is 10^-2*(B-A)/2, one hundredth of the IPOP setting, so their reported comparison does not isolate population growth as the only changed factor.

⚠️ Curated Limitations & Negative Results

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes
AIO Target Class Mapping Relation Mapping Rationale
AIO:ExponentialFunction
Exponential Function
related_match The original named whole algorithm fixes a CMA-ES base whose mandatory cumulative step-size adaptation uses the scalar exponential exp. This inherited component supports the related mapping; it is not the novel population-restart rule. The algorithm is neither equivalent to nor a subclass of the scalar function, and this does not assert a neural activation, matrix elementwise exponential or exponential requirement for arbitrary host-independent restart controllers.

Evidence Sources & Primary Literature

4 Sources
Source ID Publication / Document Type / Access Reference & Checksum
auger2005ipop
Anne Auger, Nikolaus Hansen
primary paper
full_text
ID: https://doi.org/10.1109/CEC.2005.1554902
sha256:67e91cca499d…
hansen2004
Nikolaus Hansen, Stefan Kern
primary paper
full_text
sha256:81fb79a0bc91…
auger2005local
Anne Auger, Nikolaus Hansen
primary paper
full_text
sha256:caefccd2984e…
ipop-author-errata
Anne Auger, Nikolaus Hansen
official documentation
documentation
sha256:addac47190bc…

Curation Provenance

Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-15T06:13:45Z
Updated: 2026-10-15T06:13:45Z
Source File: data/curated/IPOPCMAES.yaml