SLIM: Simplex-Lattice Interpolation Merging

Can large language models with substantially different parameter spaces be merged by direct weighted averaging, without training or semantic alignment? Existing heterogeneous fusion methods typically introduce distillation, adapters, learned latent spaces, routing, or feature alignment, leaving open whether a simpler recipe can work for genuinely different billion-parameter checkpoints. We revisit this counterintuitive question through training-free dimensional adaptation followed by ratio-controlled interpolation. In union-style merging, we expand the smaller model into the larger parameter space; in intersection-style merging, we truncate the larger model into the smaller parameter space. Across Qwen-family model pairs and benchmarks covering mathematical reasoning, code generation, language understanding, commonsense reasoning, knowledge, and instruction following, deterministic expansion largely preserves the source model function, and small-ratio interpolation can improve over strong source checkpoints by transferring complementary capabilities. However, near-balanced interpolation often collapses, and task-level results reveal a seesaw effect in which gains on some capabilities coexist with regressions on others. These results show that simple parameter averaging, when paired with lightweight dimensional adaptation and carefully controlled ratios, is a surprisingly strong baseline for heterogeneous LLM merging, suggesting that the limits of direct weighted fusion may also bound what more complex heterogeneous merging methods can achieve at scale.

Multi-task model merging combines separately trained expert models into a single model that handles all tasks without co-training. Standard practice merges experts at their optimal validation loss. We challenge this convention by systematically studying how training duration of domain experts affects the quality of the merged model. We fine-tune experts on five domains (Math, Code, Instruction Following, Multilingual, and Safety) across three model sizes (Qwen 3.5 0.8B, 2B, and 4B), saving checkpoints from 25% to 500% of the optimal training steps and evaluating five merging methods at each duration. Our findings reveal a striking method-dependent pattern: simple averaging degrades sharply with overfitting, while sparsification-based methods achieve their best performance well past the validation optimum. We formalize this through bias-variance decomposition analysis, drawing a parallel to random forests where averaging benefits from high-variance individual learners. These results suggest that training duration and merging method should be chosen jointly rather than independently.

Model merging has emerged as a lightweight paradigm for enhancing Large Language Models (LLMs), yet its underlying mechanisms remain poorly understood. In this work, we analyze late-stage pre-training trajectories and uncover a \textbf{Rank-1 Subspace} phenomenon: while raw optimization steps oscillate violently, consecutive \emph{merged} checkpoints collapse onto a stable, approximately one-dimensional linear manifold. We theoretically ground this observation in a \emph{river-valley} landscape analysis: averaging acts as a geometric low-pass filter that dampens high-curvature noise to reveal the optimal descent direction. Capitalizing on this insight, we propose \textbf{Extra-Merge}, a training-free strategy that extrapolates along this subspace to minimize loss without additional gradient updates. Extensive experiments across GPT-2 and LLaMA families (124M to 2B) demonstrate that Extra-Merge consistently outperforms standard merging baselines. Notably, it yields consistent zero-shot accuracy gains on Pythia-12B downstream tasks and generalizes effectively to the Muon optimizer \citep{jordan2024muon}.