Cooperative Multi-population GP (MEGP)

• Cooperative Multi-population Genetic Programming (MEGP) is a framework that partitions feature space into distinct views and evolves specialized GP subpopulations, combining them through an ensemble-based fitness mechanism.
• It employs Semantic-Preserving Feature Partitioning (SPFP) and soft cooperative coevolution to ensure each population specializes on different feature subsets while contributing complementarily to the final classifier.
• MEGP uses dual-level fitness evaluation—isolated and ensemble fitness—to enhance diversity retention, reduce premature convergence, and achieve superior classification performance on complex, high-dimensional datasets.

Cooperative Multi-population Genetic Program, introduced as Multi-population Ensemble Genetic Programming (MEGP), is a genetic programming framework for classification in high-dimensional, heterogeneous feature spaces that combines cooperative coevolution, multi-view learning, and ensemble learning within a single evolutionary architecture (Khorshidi et al., 16 Sep 2025). The defining idea is to partition the input space into feature views, evolve a distinct GP subpopulation on each view, and couple those otherwise isolated populations through an ensemble-based fitness mechanism rather than through migration or inter-population crossover. In this formulation, each individual is itself a multi-gene symbolic model with a differentiable softmax-based aggregation layer, while the final classifier is an ensemble across populations. The result is a dual-level evolutionary system in which specialization is enforced by feature-space decomposition and cooperation is induced by shared ensemble performance.

1. Conceptual definition and architectural principles

MEGP operates through three coupled stages: feature partitioning, independent GP evolution per view, and cross-population ensembling with cooperative fitness (Khorshidi et al., 16 Sep 2025). Formally, an input space $\mathcal{X} \subset \mathbb{R}^d$ is partitioned into $V$ views $\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$, each population $P^{(v)}$ evolves using only features in its associated view, and the resulting per-population outputs are linearly combined into a global ensemble. This makes MEGP a cooperative multi-population genetic program in a precise sense: populations remain structurally isolated in feature space, but they are rewarded partly by how well they contribute to a joint classifier.

The cooperative mechanism is explicitly described as soft cooperative coevolution. Genetic operations occur only within populations; there is no feature leakage, no inter-population crossover, and no explicit migration (Khorshidi et al., 16 Sep 2025). Cooperation is therefore not encoded in representation mixing but in credit assignment. An individual can survive because it performs well on its own view or because it combines well with individuals from other views in the best ensemble. This distinction is central to the framework and separates MEGP from classic island-model interpretations of multipopulation evolutionary search.

This design places MEGP within the broader class of multi-population evolutionary systems for which explicit architectural notation can be useful. The mpEAd formalism was proposed precisely to represent population nodes, computation nodes, and the flow of genetic and evaluative information in such systems (Lenartowicz et al., 2016). In mpEAd terms, a cooperative MEGP is naturally expressed as multiple GP population nodes feeding a joint evaluation or combination node, with evaluative feedback distributed back across populations. That correspondence does not define MEGP, but it clarifies its position inside the wider theory of multi-population evolutionary computation.

2. Multi-view decomposition and feature-space specialization

A central premise of MEGP is that high-dimensional classification becomes more tractable when the feature space is decomposed into complementary views (Khorshidi et al., 16 Sep 2025). The framework uses Semantic-Preserving Feature Partitioning (SPFP) to build feature subsets that are complementary, low-redundancy, and semantically coherent. Operationally, SPFP analyzes feature relevance, examines inter-feature dependencies and correlations, and partitions the feature space so that each view remains predictive while avoiding redundant overlap across views.

The idealized target for such a partition is conditional independence across views given the label,

$$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$

although the construction is explicitly approximate rather than exact (Khorshidi et al., 16 Sep 2025). The point is not to enforce a strict probabilistic factorization, but to obtain subsets with complementary signal so that different populations can specialize on different aspects of the prediction task.

This decomposition is motivated by several well-known pathologies of single-population GP in large feature spaces. A single GP population operating on all $d$ dimensions must search a substantially larger hypothesis space, tends to waste effort on redundant or weakly informative variables, and is more vulnerable to overfitting and premature convergence (Khorshidi et al., 16 Sep 2025). By splitting the original space into $V$ views of size approximately $d/V$, MEGP reduces the per-population search burden while preserving the possibility of recovering global performance through ensembling.

In the reported experiments, SPFP was used to produce $V=2$ views per dataset (Khorshidi et al., 16 Sep 2025). The paper also allows random partitioning as a comparison or ablation condition, following earlier multi-view GP practice, and notes that such random decomposition can work surprisingly well while being less computationally intensive. A plausible implication is that some of MEGP’s gains derive from decomposition itself, while SPFP attempts to make that decomposition semantically aligned with the task.

3. Population structure, symbolic representation, and prediction mechanism

The experimental MEGP configuration uses two populations of 30 individuals each, for a total of 60 individuals, matching the size of the single-population baseline GP (Khorshidi et al., 16 Sep 2025). Evolution proceeds in parallel for up to 150 generations, with early stopping when fitness stalls for 30 generations. Each population is tied to a distinct feature subset and evolves only within that restricted terminal space.

Each individual is a multi-gene tree-based GP model. The reported configuration uses 10 genes per individual, maximum tree depth 10, arithmetic operators $\{+, -, \times, /\}$, and terminals consisting of view-specific features plus ephemeral constants in $V$0 (Khorshidi et al., 16 Sep 2025). For an individual $V$1 in population $V$2, the gene outputs $V$3 are aggregated into class logits through a linear layer,

$V$4

followed by softmax probabilities

$V$5

The symbolic structure of the genes is evolved, while the weights $V$6 and biases $V$7 are optimized by mini-batch SGD with batch size $V$8, 1000 epochs, and learning rate $V$9 (Khorshidi et al., 16 Sep 2025).

This makes each GP individual a hybrid symbolic-parameterized model rather than a purely symbolic tree. The within-individual softmax layer serves two roles. First, it turns the multi-gene representation into a probabilistic classifier. Second, it provides interpretable gene-level coefficients that indicate how strongly each symbolic component contributes to the final decision.

Across populations, MEGP defines ensemble probabilities through a linear combination of population outputs,

$\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$0

where $\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$1 is a population-level ensemble weight for class $\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$2 (Khorshidi et al., 16 Sep 2025). These ensemble weights are optimized by minimizing mean squared error between one-hot labels and ensemble probabilities, using COBYLA. The two-level weighting structure yields a clear decomposition of responsibility: gene-level weights encode intra-individual contributions, and population-level weights encode inter-view contributions.

A closely related use of cooperative multi-population GP appears in symbolic surrogate modeling for frozen Transformer embeddings, where disjoint SPFP views are learned over embeddings and per-view symbolic programs are added to form class logits (Khorshidi et al., 16 Sep 2025). In that setting, the final model has the additive form

$\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$3

which preserves the same view-wise modularity while emphasizing interpretable closed-form logit programs.

4. Fitness, hybrid selection, and the cooperative coevolution mechanism

MEGP uses two fitness signals: isolated fitness and ensemble fitness (Khorshidi et al., 16 Sep 2025). Isolated fitness measures how well an individual performs within its own view, using cross-entropy or log-loss on the per-population softmax output. Ensemble fitness measures how well individuals perform when combined across populations into a global ensemble, again using cross-entropy on the ensemble prediction. Lower values are better in both cases.

The practical consequence is that credit assignment is deliberately non-local. For each run, candidate ensembles are formed using individuals from all populations, ensemble weights are optimized, and the individuals that participate in the best ensemble receive high ensemble-fitness credit (Khorshidi et al., 16 Sep 2025). The paper emphasizes that ensemble-best individuals do not always have the best isolated fitness; they are selected for complementary contributions. This directly encodes the intuition that cooperative ensembles need not be composed of the strongest stand-alone classifiers.

Selection combines these two signals through dual elitism and dual-criterion tournaments. A fraction $\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$4 of individuals is preserved according to isolated fitness, and a fraction $\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$5 according to ensemble fitness contributions. Tournament selection is controlled by a parameter $\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$6, which determines whether a tournament is guided by ensemble fitness or isolated fitness (Khorshidi et al., 16 Sep 2025). Five configurations are defined:

• MEGP$\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$7: $\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$8, $\mathcal{X}^{(1)},\dots,\mathcal{X}^{(V)}$9
• MEGP$P^{(v)}$0: $P^{(v)}$1, $P^{(v)}$2
• MEGP$P^{(v)}$3: $P^{(v)}$4, $P^{(v)}$5
• MEGP$P^{(v)}$6: $P^{(v)}$7, $P^{(v)}$8
• MEGP$P^{(v)}$9: $$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$0, $$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$1

This hybrid credit-assignment scheme is the operational core of MEGP’s cooperative coevolution. It preserves strong specialists within each view while also promoting individuals that may be mediocre in isolation but highly effective in combination. The resulting search dynamic broadens the set of reproductively valuable individuals and is explicitly intended to reduce premature convergence (Khorshidi et al., 16 Sep 2025).

A common misconception is to equate any multi-population evolutionary system with an island model. MEGP differs from migration-centric multipopulation schemes in a fundamental way. In island-model MPGA, populations are connected by an adjacency matrix and periodically exchange individuals, so cooperation or information flow is mediated by migration over a network (Messias et al., 2018). MEGP does not use that mechanism. Its populations remain feature-invariant, and cooperation is embodied in the ensemble loss landscape rather than in physical movement of genotypes between islands.

5. Evolutionary dynamics, benchmarks, and empirical results

The reported evaluation covers eight benchmark datasets: APSF, ARWPM, GECR, GFE, GSAD, HAPT, ISOLET, and PD, spanning 129 to 20,531 features and 2 to 26 classes (Khorshidi et al., 16 Sep 2025). The baseline comparator is a single-population GP model with 60 individuals, the same multi-gene representation, the same function set and hyperparameters, no feature partitioning, and no ensemble across populations. Each dataset is split into 54% training, 13% validation, and 33% test, and results are reported over 30 independent runs.

The empirical pattern is consistent across several axes. On challenging datasets such as HAPT, ISOLET, GSAD, and GFE, MEGP configurations—especially MEGP$$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$2 through MEGP$$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$3—converge faster and reach lower log-loss than the baseline (Khorshidi et al., 16 Sep 2025). On simpler datasets such as APSF and GECR, both methods achieve very low log-loss, around $$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$4, so the performance gap is correspondingly small. This matters because it indicates that the cooperative architecture is most beneficial where feature dimensionality, heterogeneity, or class structure make the search problem difficult.

The paper distinguishes several convergence-related measures. For Crossover Convergence Rate (CCR), Friedman tests with Bonferroni adjustment yield highly significant $$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$5-values, $$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$6, and MEGP$$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$7, MEGP$$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$8, and MEGP$$X^{(v)} \perp X^{(v')} \mid Y \quad \text{for } v \neq v',$$9 consistently outperform the baseline across generational intervals and overall (Khorshidi et al., 16 Sep 2025). By contrast, Convergence Rate (CR) is more nuanced: the baseline can sometimes show higher $d$0 because it starts from worse initial fitness and therefore registers larger absolute change. The paper explicitly warns that smaller CR does not imply inferior optimization when final solutions are better.

Diversity retention is one of the strongest reported effects. Final-population entropy is significantly higher in MEGP, especially in MEGP$d$1 to MEGP$d$2, with Friedman $d$3-values much smaller than $d$4, and the stronger cooperative variants win on 7 of 8 datasets with no losses in the reported win-tie-loss table (Khorshidi et al., 16 Sep 2025). Ranking plots further show that ensemble-best individuals often come from mid-range isolated ranks, which supports the interpretation that MEGP exploits complementary diversity rather than simply amplifying the best single-view performers.

Generalization metrics on test data also favor the cooperative design. Log-loss wins occur on 4 to 6 datasets depending on configuration and never become losses; Precision, Recall, and $d$5 show 4 wins, 4 ties, and 0 losses per configuration; AUC also shows 4 wins, 4 ties, and 0 losses (Khorshidi et al., 16 Sep 2025). Improvements are especially pronounced on multi-class, high-dimensional datasets such as HAPT and ISOLET. Cliff’s $d$6 effect sizes are reported as mostly medium-to-large positive dominance for MEGP over the baseline on these metrics.

The principal trade-off is computational cost. MEGP is consistently slower than the baseline GP because of SPFP partitioning, gradient-based softmax training per individual, and ensemble-weight optimization (Khorshidi et al., 16 Sep 2025). The paper nevertheless argues that, despite this overhead, MEGP scales better in search quality and often in wall-clock convergence because it achieves faster fitness improvement per generation.

6. Interpretability, nomenclature, related methods, and future directions

MEGP is explicitly positioned as an interpretable evolutionary classifier (Khorshidi et al., 16 Sep 2025). The symbolic genes are inspectable expression trees, gene-level weights $d$7 quantify within-individual importance, and population-level weights $d$8 indicate which views drive each class decision. SPFP views are themselves interpretable as semantic clusters such as sensor groups or gene clusters. This permits analyses of which symbolic trees matter, which views dominate particular classes, and how misclassifications arise from view-level probability contributions.

The framework is also described as structurally adaptive. Gene structures change over generations, the composition of the best ensemble changes over time, and the relative importance of isolated versus ensemble selection is tunable through $d$9 (Khorshidi et al., 16 Sep 2025). MEGP$V$0 emphasizes more independent view-based learning, whereas MEGP$V$1 to MEGP$V$2 embody increasingly ensemble-driven evolution. The paper identifies dynamic view generation, adaptive population sizing, advanced credit assignment, multi-objective MEGP, neural-GP hybrids, and distributed or streaming scenarios as future directions.

Within the surrounding literature, MEGP is framed as an extension of multi-population GP, cooperative coevolution, ensemble GP, and multi-view GP (Khorshidi et al., 16 Sep 2025). A related multi-level framework, cMLSGA, similarly distinguishes between individual-level and collective-level fitness, promotes sparse inter-population communication, and treats diversity of co-evolutionary strategies as more important than the performance of any single embedded algorithm (Grudniewski et al., 2021). This suggests a broader methodological pattern in which population-level structure is used not merely for parallelism but for controlled specialization and search diversification.

The acronym MEGP is not unique in the literature. In a separate regression setting, MEGP denotes multi-equation GP, where repeated GP runs produce multiple equations associated with clusters of training data and cooperation occurs at prediction time through cluster- and distance-weighted combination (Ghasemi et al., 2023). That usage is ensemble-like but not a cooperative multi-population evolutionary process, because the GP runs are independent and sequential rather than jointly coevolved. Distinguishing these two meanings is essential when comparing papers.

A final point of clarification concerns the scope of cooperation. In cooperative multi-population GP, cooperation can be represented diagrammatically as the assembly and joint evaluation of individuals from multiple populations (Lenartowicz et al., 2016), or theoretically contrasted with migration-based island dynamics (Messias et al., 2018). MEGP’s contribution is to realize cooperation through feature-view decomposition and ensemble-level fitness rather than through explicit migration, while preserving symbolic transparency. This combination of modularity, dual-level credit assignment, and interpretable probabilistic outputs defines the framework’s distinctive place in evolutionary machine learning.