MERGE^3: Efficient Evolutionary Neural Merging

• MERGE^3 is an evolutionary framework that integrates evolutionary search with IRT-based performance estimation to merge neural network weights without further fine-tuning.
• It employs aggressive evaluation subsampling, reducing fitness evaluation costs up to 50-fold while maintaining competitive performance on multi-task and multilingual benchmarks.
• The framework uses genetic operators like SBX and Gaussian mutation to optimize weight interpolation, achieving high-quality model merges with reduced computational expense.

MERGE$^3$ is an efficient evolutionary model merging framework for neural networks that integrates evolutionary search in weight space with Item Response Theory (IRT)-based performance estimation to enable high-quality multi-task and multilingual model merging on consumer-grade GPUs. The framework achieves up to a 50-fold reduction in fitness evaluation costs compared to traditional evolutionary merging, while preserving empirical performance and solution quality compared to full-dataset evaluations. MERGE$^3$ is the only approach in its class to combine aggressive evaluation subsampling, statistically principled performance proxies, and evolutionary operators into a unified system for black-box model merging without further fine-tuning or access to original training data (Mencattini et al., 9 Feb 2025, Crisostomi, 2 May 2026).

1. Conceptual Motivation and Problem Setting

Model merging refers to the composition of several independently trained neural networks—typically with disjoint or specialized task competencies—into a single unified model directly at the parameter (weight) level. This paradigm enables the reuse, composition, and transfer of learned capabilities outside the standard supervised or continual learning pipeline, with applications in multi-task transfer, cross-lingual adaptation, and modular system design (Crisostomi, 2 May 2026).

Evolutionary merging is particularly suited for discovering nontrivial combinations of model weights, bypassing the limitations of deterministic weight-averaging or low-rank merges. However, the main bottleneck in evolutionary methods is the computational expense: each candidate solution (merged model) must ordinarily be evaluated against the full validation set, with the total number of forward passes scaling linearly in both candidate count and dataset size. MERGE$^3$ is designed to overcome this limitation, making population-based model merging feasible even on resource-constrained hardware (Mencattini et al., 9 Feb 2025).

2. Framework Architecture and Methodology

MERGE$^3$ is structured around three principal stages, each contributing to substantial computational gains:

1. Extraction (Extract): Given a large evaluation dataset $D$, MERGE$^3$ samples a small, uniformly drawn subset $\bar{D}$ (typically $2$–$5\%$ of $|D|$), which is used as a calibration set for candidate evaluation. This subset size is traded off against expected error in fitness estimation (Mencattini et al., 9 Feb 2025).
2. Estimation (Estimate): MERGE$^3$0 fits a multidimensional 2-parameter logistic IRT model to the full data, yielding item discrimination $^3$1, difficulty $^3$2, and model ability vectors $^3$3 for each endpoint model $^3$4. Linear inheritance is assumed: the ability of a merged model parameterized by weights $^3$5 is $^3$6. For a given merge, performance on hold-out items in $^3$7 is efficiently estimated as $^3$8 (Crisostomi, 2 May 2026).
3. Evolution (Evolve): Candidate merges are represented either as interpolation coefficients $^3$9 on the simplex or as full genetic encodings. Evolution uses selection (tournament or Pareto for multi-objective search), simulated binary crossover (SBX) applied per layer or parameter block, and Gaussian mutation. Fitness for each candidate is computed using the mp-IRT estimator:

$^3$0

where $^3$1 and $^3$2 (Mencattini et al., 9 Feb 2025).

The core workflow is summarized below:

Stage	Operation	Technical Principle
Extraction	Subsample calibration set $^3$3	Uniform sampling
Estimation	IRT ability inference, performance prediction	2PL IRT, linear inheritance
Evolution	Candidate population update, fitness computation	Genetic EA, SBX, mp-IRT estimator

3. Mathematical Formulation and Theoretical Guarantees

Let $^3$4 denote model $^3$5’s correctness on item $^3$6. The multidimensional 2PL IRT likelihood for a model’s performance on item $^3$7 is

$^3$8

with $^3$9.

For a merge with coefficients $^3$0 (simplex), the merged ability $^3$1 allows prediction of success on all examples in $^3$2 by

$^3$3

The mp-IRT estimator aggregates ground-truth observations on $^3$4 with IRT-predicted correctness on $^3$5. Generalized estimator variants (gmp-IRT) add convex weighing to reduce bias.

Theoretical Properties

• $^3$6-Stability: If the score $^3$7 computed on $^3$8 is $^3$9-stable with respect to the true score $D$0, the optimal candidate selected on $D$1 is within $D$2 of the global optimum (Theorem 1, (Mencattini et al., 9 Feb 2025)).
• Sample Complexity Bound: The estimation variance decays as $D$3 for $D$4 items in $D$5 (Estimation-Variance Bound, (Crisostomi, 2 May 2026)).
• Empirical Efficiency: Using $D$6 samples for datasets with $D$7–$D$8 achieves $D$9 reduction in forward passes with negligible accuracy loss.

4. Empirical Validation and Benchmarks

MERGE$^3$0 has been validated on diverse tasks and modalities:

• Multilingual and Cross-Lingual Merging: Merging math-tuned English models with Italian, German, Dutch, and Japanese Mistral-7B variants on translated GSM8K achieves $^3$1–$^3$2 absolute gains over endpoints, outperforming Task Arithmetic, SLERP, TIES, and DARE variants.
• Multi-Task ARC Benchmark: Multi-objective evolutionary merging on translated ARC with four monolingual models results in $^3$3 to $^3$4 improvement over endpoints using only $^3$5 samples per language for fitness evaluation.
• Comparative Baselines: MERGE$^3$6 with IRT-based evaluation matches or slightly trails full-dataset evolutionary search (e.g., $^3$7 vs. $^3$8 average accuracy on vision/language tasks) but at $^3$9 versus $\bar{D}$0 forward passes.
• Speed and Feasibility: On a single NVIDIA 4090, MERGE$\bar{D}$1 with $\bar{D}$2 processes approximately $\bar{D}$3 models/hour; with $\bar{D}$4–$\bar{D}$5, $\bar{D}$6–$\bar{D}$7 models/hour is feasible, enabling practical evolutionary searches on desktop-class GPUs.

5. Algorithmic Procedure and Implementation

A high-level pseudocode instantiates MERGE$\bar{D}$8 as follows (Crisostomi, 2 May 2026):

1. Fit IRT item parameters $\bar{D}$9 on a model pilot set.
2. Initialize population by random interpolation between endpoint models.
3. For $2$0 generations:
   • For each candidate, evaluate on $2$1, estimate $2$2 by IRT-MLE, compute $2$3.
   • Select elites via tournament, create offspring with SBX and mutation.
4. Periodically recalibrate the affine mapping between $2$4 and $2$5 by full-dataset evaluation to avoid estimator drift.
5. Return Pareto-optimal set (multi-objective) or the best candidate by estimated fitness.

Essential Hyperparameters

• Population size $2$6 (e.g., $2$7), generations $2$8 (e.g., $2$9)
• SBX index $5\%$0, mutation standard deviation $5\%$1, probability $5\%$2
• Calibration size $5\%$3 (recommended); recalibration every $5\%$4 generations

6. Limitations, Extensions, and Future Directions

Several intrinsic limitations are acknowledged:

• IRT capacities rely on representative pilot data for item fitting; robustness degrades if the test domain drifts significantly.
• Calibration of the affine map between IRT ability and true performance requires intermittent grounding via full-dataset evaluation.
• Merging a model with itself yields no gains, rejecting trivial "prompt-fitting" or accidental self-improvement.

Possible extensions discussed include adaptive item selection based on discriminatory power, extension to three-parameter IRT models, and hybridization with gradient-based local search methods (e.g., CMA-ES).

7. Significance and Comparison to Related Approaches

MERGE$5\%$5 occupies a distinct position in the landscape of model merging methodologies:

• Vs. Closed-Form Averaging: Goes beyond deterministic weight-averaging (e.g., "Model Soup") by enabling stochastic and non-convex interpolations through evolutionary operators.
• Vs. Task Arithmetic/Low-Rank Merging: Identifies candidate merges where analytical solutions are infeasible and exploits potential synergistic effects missed by linear subspace techniques.
• Vs. Full-Evaluation Evolution: Achieves matching solution quality at $5\%$6 lower compute cost by leveraging IRT proxies.
• Multi-Objective Capabilities: Multi-task merges supporting Pareto optimization across divergent evaluation objectives.

MERGE$5\%$7 thus provides an algorithmically principled, empirically validated, and resource-efficient pipeline for merging large neural models in settings where closed-form solutions are suboptimal or inapplicable (Mencattini et al., 9 Feb 2025, Mencattini et al., 9 Feb 2025, Crisostomi, 2 May 2026).