Subspace Control: Turning Constrained Model Steering into Controllable Spectral Optimization

Open-weight foundation models can be fine-tuned for harmful purposes after release, yet no existing training resistance methods provide theoretical guarantees. Treating these interventions as convergence-rate control problems allows us to connect optimization speed to the spectral structure of model weights. We leverage this insight to develop a novel understanding of convergence rate control through spectral reparameterization and derive an algorithm, SpecDef, that can both provably and empirically slow first- and second-order optimization in non-adversarial settings. In adversarial settings, we establish a fundamental limit on a broad class of convergence rate control methods including our own: an attacker with sufficient knowledge can restore fast convergence at a linear increase in model size. In order to overcome this limitation, future works will need to investigate methods that are not equivalent to controlling convergence rate.

Continual learning (CL) and model merging (MM) both aim to obtain a single model that performs well across multiple tasks, challenged respectively by catastrophic forgetting and weight-disentanglement error. In the literature, these difficulties are merely treated separately and mitigated through a variety of solutions, while the geometry induced by the base optimizer is treated as an implementation detail. In this work, we show that the two difficulties are in fact two instances of the same phenomenon: a parameter update useful for one task shifts the model's outputs on another. We formalize this shared phenomenon as task interference and reduce it to a common layer-wise Frobenius inner product $\langle ΔW_\ell, J_\ell(x)\rangle_F$. This quantity, in turn, is utilized to expose the role of the optimizer. We theoretically derive an upper bound that isolates the spectral norm $\|ΔW_\ell\|_2$ as an optimizer-controllable factor of task interference, and a per-mode analysis shows that this bound tracks the dominant part of the empirical interference. Specifically, we then identify the recent Muon optimizer as a mechanism that regulates this factor by construction. Our work reveals that its elegant control on spectral norm tightens the interference bound for both CL and MM, positioning Muon as a principled optimizer-centric approach complementary to existing solutions. Our theoretcal analysis is well validated by experimental results. Replacing the AdamW optimizer with Muon improves accuracy by up to +5.02 points on the eight-task model-merging benchmark across three CLIP backbones. For continual learning, Muon also delivers uniformly positive gains across ten class-incremental protocols, three task-incremental protocols, and the 11-task MTIL benchmark.

While spectral-based optimizers like Muon operate directly on the spectrum of updates, standard adaptive methods such as AdamW do not account for the spectral structure of weights and gradients, leaving them vulnerable to two empirical issues in large language model (LLM) training: (i) the optimizer updates can have large spectral norms, potentially destabilizing training and degrading generalization; (ii) stochastic gradient noise can exhibit sparse spectral spikes, with a few dominant singular values much larger than the rest. We propose SPECTRA, a general framework addressing these by (i) post-spectral clipping of updates to enforce spectral-norm constraints (ii) optional pre-spectral clipping of gradients to suppress spectral noise spikes. We prove that post-clipping constitutes a Composite Frank-Wolfe method with spectral-norm constraints and weight regularization. We further analyze how pre-clipping mitigates sparse spectral spikes. We propose efficient soft spectral clipping via Newton-Schulz iterations, avoiding expensive SVD. Experiments on LLM pretraining show SPECTRA uniformly improves validation loss for various optimizers, including AdamW, Signum, Mars, and AdEMAMix, with the best-performing variants achieving state-of-the-art results. Models trained with SPECTRA exhibit smaller weight norms, confirming the link between spectral clipping and regularization.