Iterative Merging Scheme Overview

• Iterative Merging Scheme is a bottom-up algorithm that progressively combines atomic units via guided merge operations based on optimality criteria.
• It underpins diverse applications in optimization, machine learning, decentralized systems, and VLSI design by facilitating adaptive model fusion and clustering.
• The method's mathematical underpinnings, including submodular optimization and MCMC, offer theoretical guarantees and efficient convergence.

An iterative merging scheme is a general algorithmic principle in which a collection of basic entities (such as models, parameters, data, modules, or nodes) are progressively combined through a sequence of merge operations, typically with a well-defined local or global optimality criterion. These schemes are pervasive across optimization, machine learning, distributed systems, circuit design, geometric approximation, and combinatorial problems. In their modern incarnations, iterative merging methods are central in multi-model fusion, federated and decentralized learning, hierarchical clustering, cooperative data exchange, floorplanning, Markov chain Monte Carlo sampling, and data reduction. This article surveys the mathematical foundations, canonical algorithmic structures, theoretical guarantees, and selected advanced applications, with a focus on rigorously specified instances from the research literature.

1. Canonical Paradigms and Structural Overview

Most iterative merging schemes comprise three key ingredients: a set of initial atomic elements; a pairwise or multi-way merge operator, guided by a (problem-dependent) cost function or merging criterion; and an update or coordination strategy deciding which elements to merge at each iteration. Essential characteristics distinguishing iterative merging from one-shot or batch approaches are locality, feedback, and multi-step optimization.

The archetypal sequence is:

1. Initialization: Start with singletons or minimally structured units (e.g., per-task models, individual client states, single-vertex partitions).
2. Merge Selection: Repeatedly identify a pair or set of units whose merge greedily or optimally reduces cost, distortion, rate, or some alternative surrogate.
3. Merging Step: Combine these units via an analytic, combinatorial, or data-driven operator (e.g., averaging, optimal alignment, joint re-optimization).
4. Bookkeeping and Feedback: Update the population; record any necessary auxiliary structure (e.g., merge history, slicing trees, masks, schedules); adjust selection or stopping rules as needed.
5. Iteration/Stabilization: Repeat until a global structure emerges, desired number of entities remains, or convergence is proven.

This schema encompasses greedy, submodular, Markov chain Monte Carlo, data-driven, or decentralized variants, each instantiating different merge operators and control strategies. In contrast to recursive, top-down, or divide-and-conquer arrangements (e.g., recursive sorts or splits), iterative merging is fundamentally a bottom-up, constructive process.

2. Theoretical Foundations and Algorithmic Guarantees

Iterative merging schemes are mathematically justified via submodular optimization, dynamic programming, convex and nonconvex minimization, local optimality, Markov chain theory, or information-theoretic bounds, depending on context.

• Submodular Optimization: Merging partitions or coalitions can often be formulated as a minimization over a submodular set function, leading to polynomial-time algorithms with guaranteed optimality under certain conditions (Ding et al., 2015).
• Greedy and Local Optimality: Algorithms such as minimal-area iterative merging in VLSI floorplanning (He et al., 2014) or minimal-error point deletion in polygonal approximation (Ray, 5 Jun 2025) exploit greedy selection at each step, with cumulative global improvement.
• Gradient and Loss Surrogates: In model merging, task vectors (parameter deltas) are often shown (via Taylor expansion or direct calculation) to approximate or realize the gradient of a joint objective, so iterative merging steps mathematically relate to multi-step gradient descent (Zhou et al., 5 Nov 2024).
• Metropolis–Hastings and MCMC: In statistical inference, iterative merge–split moves are formalized as proposal kernels in the Metropolis–Hastings Markov chain, with ergodic mixing and detailed balance guarantees (Peixoto, 2020).
• Convergence and Error Bounds: For decentralized learning and federated optimization, iterative merging is proven to achieve minimax consensus error, with quantifiable dependence on spectral properties of the communication topology (Saadati et al., 11 Apr 2024).

Traditionally, correctness arguments involve establishing that each merging step preserves or reduces a global cost (e.g., error, entropy, rate) and that the series of merges reaches a terminal, optimal, or near-optimal state. In advanced scenarios, constraints such as privacy, resource constraints, or data access restrictions dictate the choice and analysis of merging protocols.

3. Canonical Instances and Applications

3.1 Model and Adapter Merging

• Alternating Tuning and Merging (ATM): ATM (Zhou et al., 5 Nov 2024) iteratively alternates per-task fine-tuning (yielding task vectors) and a merging phase (aggregating updates into a base), repeatedly. ATM's update at each iteration is provably equivalent to a multi-task joint gradient step, and repeated alternations empirically and theoretically yield improved joint performance both in standard and federated settings.
• Optimal Brain Iterative Merging (OBIM): OBIM (Wang et al., 17 Feb 2025) combines saliency-based parameter pruning (measuring the marginal loss impact of each weight) with an exclusive-masking merge step, yielding merged models that avoid both intra-model and inter-model destructive interference, outperforming prior approaches on both general and cross-lingual LLM fusion tasks.
• Iterative Inference-Solving Alignment (IterIS): IterIS (Chen et al., 21 Nov 2024) for LoRA merging efficiently solves for a single unified adapter by iterating inference (extracting features with the current merged adapter) and regularized closed-form least squares, provably requiring only a small number of iterations and minimal samples for stable convergence.
• Particle Swarm Optimization Merging (PSO-Merging): PSO-Merging (Zhang et al., 27 Aug 2025) interprets entire expert models as particles, propagating them through standard PSO velocity updates. The true merged model is selected as the best performer along the optimization trajectory. This approach achieves state-of-the-art results for multitask LLM fusion, outperforming both gradient-based and classical data-free algorithms.

3.2 Cooperative Data Exchange

• Iterative Merging Algorithm (IM): In non-packet-splitting cooperative exchange systems, IM (Ding et al., 2015) finds the minimum sum-rate strategy by recursively merging coalitions whose joint local recovery reduces an explicit deficit over a submodular set function. At each step, the algorithm greedily identifies a minimizer among possible merges, achieving provable optimality and substantial runtime improvements for moderate system sizes. See the table below.

Step	Description	Reference
Merge selection	Subset of coalitions with maximal deficit decrease	(Ding et al., 2015)
Local rate allocation	Cut-set bound-based allocation for merged coalition	(Ding et al., 2015)
Global stopping	When only two coalitions remain, exact solution attained	(Ding et al., 2015)

3.3 Geometric Approximation and Floorplanning

• Iterative Merging Placement (IMP): In VLSI floorplanning, IMP (He et al., 2014) constructs composite modules via repeated merging of minimal-area pairs, building a binary slicing tree, then top-down placement for geometric realization. The relaxed feasibility condition on composite aspect ratio strictly improves over prior zero-deadspace solvers.
• Polygonal Approximation: In digital curve simplification, iterative merging selectively deletes vertices of minimal error (locally defined as perpendicular distance to adjacent chords), yielding a minimal-feature set with guaranteed geometric fidelity and efficient $\mathcal{O}(m \log m)$ complexity (Ray, 5 Jun 2025).

3.4 Iterative Merging in Distributed Optimization and Clustering

• DIMAT Framework: In decentralized deep learning, DIMAT (Saadati et al., 11 Apr 2024) alternates local SGD with neighbor-wise iterative activation-matching merges, aligning layers across agents and averaging parameters to provably accelerate consensus and reduce communication overhead.
• Community Detection (Merge–Split MCMC): Stochastic block model inference leverages iterative merge–split MCMC (Peixoto, 2020), in which group merges and splits are designed as proposal kernels for efficiently exploring partition space, surpassing single-node moves in mixing rate by several orders of magnitude.

4. Detailed Algorithmic Schemes

Precise algorithmic variants abound. The following archetypes are widely cited:

• Pairwise/Subset Merge Greedy (IM, IMP, Polygonal): At each step, select minimal-deficit/mistake/error pairs (or subsets), merge to form new composite unit, and update.
• Alternating Tuning and Merging (ATM): For $K$ rounds: fine-tune each task, compute task vectors, aggregate via arithmetic mean (or, in OBIM, exclusive mask-based merge), and update the base model. Iteration count and per-round granularity directly affect empirical performance.
• Particle Swarm: Each particle represents a full parameter set; at each iteration, positions and velocities are updated according to PSO rules, and the current best is retained.
• MCMC Merge–Split: Iteratively propose merges or splits of entire groups in the partition, alternate with fine-grained (e.g., single-node) moves for ergodicity. Each move is subject to the Metropolis–Hastings acceptance criterion for stationary sampling.

For many of these algorithms, formal pseudocode is available in the referenced works, including full Dafny code for iterative mergesort (Carbonell et al., 1 Sep 2025) and full proposal/acceptance kernels for MCMC merge–split on networks (Peixoto, 2020).

5. Theoretical Performance and Empirical Impact

Rigorous analysis has elucidated performance bounds, convergence properties, and practical advances of iterative merging techniques:

• Optimality: For submodular coalition merging, IM achieves the theoretical minimum sum rate for universal recovery (Ding et al., 2015).
• Acceleration: For community detection, merge–split MCMC yields decorrelation times orders of magnitude shorter than single-node methods and enables sampling even in large, complex graphs (Peixoto, 2020).
• Communication Efficiency: In decentralized learning, iterative merging reduces communication rounds by factors of $5$–$10$ without compromising model performance (Saadati et al., 11 Apr 2024).
• Multi-Task Model Quality: Repeated alternating/iterative merging in model fusion (ATM, OBIM, PSO-Merging) substantially improves joint task accuracy over one-shot or single-pass methods, with empirical gains up to $+20$ percentage points in vision tasks (Zhou et al., 5 Nov 2024, Wang et al., 17 Feb 2025, Zhang et al., 27 Aug 2025).

6. Extensions and Application Domains

Iterative merging schemes have been generalized to support:

• Hierarchical and Nested Structures: Merging can be extended to hierarchical partitions or multilayer models, as in hierarchical SBM sampling (Peixoto, 2020) or multi-stage adapter fusion.
• Adaptive and Signal-Guided Merging: Continuous monitoring of signals (learning progress, forgetting, replay buffer statistics) guides dynamic scheduling of merge events for continual learning in large LLMs (Feng et al., 22 Sep 2025).
• Decentralized or Federated Constraints: Where centralization of data or parameters is infeasible, iterative merge protocols based on exchanged updates, gradients, or activation statistics enable both privacy-preserving and communication-efficient learning (Zhou et al., 5 Nov 2024, Saadati et al., 11 Apr 2024).

7. Limitations, Open Questions, and Benchmarking

While iterative merging schemes can offer provable guarantees and empirical advances over static or batch alternatives, they remain subject to:

• Computational Overhead: Some schemes involve combinatorially many possible merges or require nontrivial oracle computation, though judicious design exploits submodularity or greedy proxies for tractability (Ding et al., 2015).
• Stability Concerns: In high-dimensional model fusion (especially via parameter averaging), naive merges can trigger destructive interference; exclusive-masking (OBIM) or signal-weighted fusion (AIMMerging) partially mitigate these effects, but theoretical convergence rates and loss bounds in the nonlinear regime remain active areas of research.
• Parameter Tuning: Hyperparameter selection (e.g., merge intervals, regularizer strengths, particle initialization) often strongly influences empirical efficacy. Recent work on adaptive and feedback-guided merging signals significant advances in robustness (Feng et al., 22 Sep 2025).

Comprehensive benchmarking shows iterative merging's superiority in several tasks:

• Model merging and multitask adaptation (Zhou et al., 5 Nov 2024, Wang et al., 17 Feb 2025, Zhang et al., 27 Aug 2025)
• Cooperative data exchange (Ding et al., 2015)
• Geometric and network partitioning (Ray, 5 Jun 2025, Peixoto, 2020)
• Floorplanning and wire-length minimization (He et al., 2014)

Ongoing work includes refining convergence theory, extending applications to new domains (e.g., streaming, non-Euclidean data), and developing principled adaptive schemes for online and nonstationary environments.

• ATM: Improving Model Merging by Alternating Tuning and Merging (Zhou et al., 5 Nov 2024)
• Iterative Merging Placement Algorithm for the Fixed-Outline Floorplanning (He et al., 2014)
• A Fast Unsupervised Scheme for Polygonal Approximation (Ray, 5 Jun 2025)
• Optimal Brain Iterative Merging: Mitigating Interference in LLM Merging (Wang et al., 17 Feb 2025)
• DIMAT: Decentralized Iterative Merging-And-Training for Deep Learning Models (Saadati et al., 11 Apr 2024)
• PSO-Merging: Merging Models Based on Particle Swarm Optimization (Zhang et al., 27 Aug 2025)
• IterIS: Iterative Inference-Solving Alignment for LoRA Merging (Chen et al., 21 Nov 2024)
• AIMMerging: Adaptive Iterative Model Merging Using Training Trajectories for LLM Continual Learning (Feng et al., 22 Sep 2025)
• Reinforcement Learning with Iterative Reasoning for Merging in Dense Traffic (Bouton et al., 2020)
• Merge-split Markov chain Monte Carlo for community detection (Peixoto, 2020)
• Faster Small-Constant-Periodic Merging Networks (Piotrów, 2014)
• Derivation and Verification of Array Sorting by Merging, and its Certification in Dafny (Carbonell et al., 1 Sep 2025)
• Iterative Merging Algorithm for Cooperative Data Exchange (Ding et al., 2015)