Merging finetuned models combines specialized capabilities without joint training or access to the original data. Most methods operate by linear arithmetic in Euclidean weight space, which cannot carry the geometry of the update. Orthogonal Model Merging (OrthoMerge) uses a single orthogonal transform for each weight matrix, but such a transform cannot change singular values. We propose CORAM, which partitions each target matrix into row slices, represents every expert slice by its singular value decomposition in the corresponding base-model SVD frame, and merges the task-specific factors on their corresponding manifolds. Because manifold averaging contracts the merged update, CORAM applies an amplification coefficient $\lambda=\kappa\hat{c}$. The scale c_hat is estimated from the expert and merged update norms and is approximately $\sqrt{N}$ for $N$ experts with comparable update magnitudes. The restoration strength kappa is selected from the dispersion of expert updates without evaluating candidate merged models. This rule remains within 0.72 points of the best swept value on all evaluated suites. CORAM also includes spread slicing to distribute highly updated rows across slices and a residual pathway for non-target layers. Across four suites covering three model families, 3B to 9B scales, and language and vision-language experts, CORAM improves over OrthoMerge by 0.25 to 1.35 points and matches or exceeds the strongest weight-space baselines.
Modern neural network training increasingly uses matrix-aware optimizers, yet their conditioned matrix step is typically added directly to the weight, jointly changing its norm and direction. This interaction matters because the current norm determines angular motion, while directional learning can drive norm growth and thereby alter later steps. We introduce RODE, which gives the radial and directional components separate update rules and step sizes. RODE explicitly updates the matrix Frobenius norm through a scalar radial rule, while its directional channel performs Newton--Schulz-conditioned updates in the tangent space. Controlled GPT-2 interventions show gains from both direct norm control and RODE's directional update. Across two language-modeling and two image-classification tasks, RODE outperforms both Muon variants in every direct comparison and ends with lower full-model norms. At 1.5B scale, using the learning rate transferred directly from the Qwen2-style LM sweep, RODE lowers loss from 4.145 to 3.346 and final global norm from 11964 to 2183 relative to Muon RMS, with fixed-radius RODE improving further. For Qwen3.5-9B full-parameter fine-tuning, all six optimizers use the same tuning budget and the same formal-training and evaluation settings; RODE outperforms both Muon variants on all four evaluation tasks and attains the highest mean on GSM8K and MATH-500. Thus, decoupling radial and directional dynamics offers a more effective and controllable approach to matrix optimization.
Weights and biases are normally optimized as separate parameter tensors, yet they do not represent separate functions when the input to an affine layer has nonzero mean. For an affine map $z=Wx+b$ with input mean $\mu$, a weight update contains a sample-independent displacement $\Delta W\mu$ that is functionally indistinguishable from a bias update. We call this hidden contribution \emph{boundary motion} and decompose each update into a centered, sample-varying \emph{shape} component and a shared \emph{boundary} component. On a four-layer Transformer trained from scratch on IMDb, the bias-like term $g_b\mu^\top$ has a median norm equal to 0.664 of the raw weight-gradient norm across affine layers and training checkpoints. More strikingly, the median ratio $\norm{\Delta W\mu}/\norm{\Delta b}$ is 134.7, while $\norm{\Delta W\mu}/\norm{\Delta b+\Delta W\mu}$ is 0.994. Thus, under AdamW, the observed boundary motion is almost entirely realized through the weight matrix rather than the explicit bias. We implement a diagnostic optimizer, Shape--Boundary Orthogonal AdamW (SBO-AdamW), that optimizes $g_W-g_b\mu^\top$ and $g_b$ with independent Adam states and compensates the weight-induced boundary displacement. In a single-seed experiment, SBO-AdamW raises validation accuracy from 81.68\% to 85.81\% and validation-selected test accuracy from 78.73\% to 82.73\%, with the best validation checkpoint occurring at step 800 instead of step 3000. However, the moving-batch-center compensation produces severe bias-coordinate drift and strongly reduces boundary energy. The present evidence therefore supports hidden boundary motion as an important optimization mechanism, but it does not yet establish a final general-purpose optimizer. A stable centered-affine parameterization is identified as the required next step.
Gongyue Zhang, Yixuan Sheng, Donghan Liu et al.· 0 citations
Learning from heterogeneous representations is often reduced to feature concatenation, erasing which representation produced each error. We propose residual algebra, in which each representation retains its coordinate system and owns its unresolved residual until an explicit aggregation boundary. Fold instantiates representations as point-in-time conditional-mean fields on 10x10 rank grids, and FPRC-PQ composes them through relax-aggregate-close: each field first fits a correction to its own residual, corrected fields then meet at a fixed mean, and a shared learner closes only the aggregate's fresh residual. We formalize aggregation as a quotient by the zero-sum redistribution kernel, characterizing legal post-aggregation operators as those constant on its cosets. The resulting composition separates representation, local residual estimation, and residual-of-residual estimation, with population variance reduction and first-order coupled-path mean orthogonality. Rumination-B and Rumination-H extend the algebra with quotient-legal finite correction and feedback. On 3.67M Chinese A-share stock-day observations (2023-2026) under a frozen point-in-time protocol, FPRC-PQ raises net-of-cost return from 13.52% to 19.10% and Sharpe from 1.42 to 2.09, outperforming matched-capacity, unified-residual, identity-free two-stage, and pairwise-only controls. The gain is thus attributable to explicit residual ownership and composition rather than additional features or trees.
We study vector-valued affine refinement operators of the form [ (W\gamma)(t)=\sum_{j\in\mathbb{Z}} A_j\gamma(Mt-j)+B(t), ] with finitely supported matrix mask and compactly supported continuous piecewise linear input and forcing data. Building on the homogeneous realization theorem for (B\equiv 0), we prove that, for (M\ge 3), every finite affine iterate (W^n\gamma) admits an exact fixed-width ReLU realization whose depth is (O(n)). The main new ingredient is a residual memory controller. It replaces the noninvertible residual dynamics by an injective skew-product and permits exact backward replay of the residual states required by a Horner-type evaluation of the affine forcing sum. Offset frames align the forcing atoms away from residual seams, allowing complementary loop readouts to recover their values exactly. The remaining branch-selection ambiguity occurs only where the accumulated affine state has already vanished. For (M\ge 3), the result applies to arbitrary compactly supported continuous piecewise linear forcing terms. For (M=2), the same construction applies to ordinary-frame seam-separated forcing. We also prove a stage-dependent extension for forcing terms in a fixed finite-dimensional continuous piecewise linear span and record the resulting linear-depth upgrade for open-curve, finite-state, and Hilbert- and Morton-type recursive constructions.
BaKron is an efficient solver that combines anti-diagonal parallelism with a recursive divide-and-conquer construction that matches the cubic scaling of GPTQ while exploiting richer curvature information.
Results show that multi-objective Bayesian optimization is valuable as a search layer for expressive merge parameterizations in model merging and introduce MOBO-Merge, a merge-operator agnostic framework that uses multi-objective Bayesian optimization to approximate the Pareto front under a limited evaluation budget.
Utkarsh Agarwal, V. Bonagiri, Raul Astudillo et al.· 0 citations
A new machine-learning framework aims to improve the success rate of computational protein design while moving away from results that reproduce sequences found in nature.
MIT News · Artificial Intelligence· news.mit.eduAug 24, 2026
