Model Merging via Data-Free Covariance Estimation

Model merging aims to combine multiple task-specific expert models into a single model without joint retraining, offering a practical alternative to multi-task learning when data access or computational budget is limited. Existing model merging methods rarely exploit strong merged models as priors for further improvement. To address this limitation, we propose ReForge, a bilevel optimization framework that formulates module-wise refinement as Bayesian linear regression with an anchor-centered prior. The inner level yields a closed-form MAP estimate from unlabeled calibration activations. The outer level uses Bayesian optimization to jointly select heterogeneous regularization strengths and assembly scales using held-out validation data. Furthermore, we develop a data-free variant of ReForge that replaces activation statistics with task-vector Grams, eliminating the need for calibration examples. Across extensive benchmarks, including up to 20-task merging in vision and 5-task merging in language, ReForge consistently outperforms all evaluated plug-and-play anchor baselines (e.g., TA, WUDI-Merging, and TSV). On 20-task ViT-B/32, ReForge improves the strongest evaluated baseline, ISO-CTS, from 77.6% to 82.8% in the data-assisted setting and to 81.5% in the data-free setting. On eight-task ViT-L/14, the data-assisted variant achieves 95.1% mean accuracy, compared with 95.8% for the individual task experts. Our source code will be released soon.

Modern deep learning usually treats models as separate artifacts: trained independently, specialized for particular purposes, and replaced when improved versions appear. This thesis studies model merging as an alternative paradigm: combining independently trained neural networks directly in weight space, with little or no optimization and without requiring access to the original training data. The thesis considers two main regimes. In the single-task setting, where models share an objective but differ in initialization, we introduce C$^2$M$^3$, a cycle-consistent merging algorithm based on Frank-Wolfe optimization. C$^2$M$^3$ aligns multiple networks into a shared, reference-free parameter space, making weight averaging meaningful without privileging any individual model. In the multi-task setting, where models are fine-tuned for different downstream tasks from a common pretrained initialization, we first develop a theoretical account of task vectors as approximate gradients. This explains both the effectiveness and the limitations of task arithmetic. Building on this view, we show that task vectors inherit the low-rank structure of gradients and introduce Task Singular Vectors (TSV), a decomposition that enables compression and interference reduction through TSV-Merge. We then present MASS, an input-adaptive routing method that uses TSV geometry to select task-relevant subspaces at inference time. Finally, we introduce MERGE$^3$, an evolutionary merging framework that uses Item Response Theory to reduce evaluation costs by up to 50$\times$ while preserving solution quality. Together, these contributions provide theoretical and algorithmic foundations for model merging, supporting a paradigm in which learned capabilities can be composed, reused, and extended across models.

Model merging combines fine-tuned checkpoints into a single multi-task model without retraining. Existing methods - such as task arithmetic, model soups, TIES, and DARE - are computationally efficient and empirically successful, but rely on heuristic design choices and lack formal optimality guarantees. We show that merging can be formulated as a convex quadratic programme over residual updates, yielding weights that minimise a squared-output calibration objective using calibration inputs and fine-tuned model outputs, and subsuming existing methods as special cases. Our framework yields a closed-form diagnostic - the fraction of residual energy captured by a chosen basis - that predicts downstream merge quality using only the calibration set. Empirically, the QP matches or outperforms existing methods in the single-layer setting, and we characterise when the optimal basis provides significant gains over the cheaper diagonal QP. We extend to multi-layer merging via a sequential layer-wise algorithm and demonstrate consistent gains across language and vision benchmarks.