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 \textit{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.
Across implicit-neural-representation (INR) architectures and analytic benchmarks we observe that a thoroughly tuned Adam (especially its learning rate (lr), e.g. in a hyperparameter sweep from $lr = 0.05$ to $10^{-8}$) can potentially reach a very low loss even on ill-conditioned loss landscape or converge at a plateau far above the loss attained by second-order methods. This report defines the measured metrics that help determine if Adam can mitigate the ill-conditioning on a given loss landscape. We provide the indicators by which each outcome is determined, that are: the condition number of the Hessian and of the Adam-preconditioned Hessian $D^{-1/2}HD^{-1/2}$ (with the derivation from Adam's update rule), the diagonal mass $ρ$ that distinguishes axis-aligned from cross-coupled ill-conditioning, the negative spectral mass estimated by stochastic Lanczos quadrature, and the gradient energy fractions over curvature bands, including the flat fraction that indicates the Adam stall. A worked out $2\times 2$ example and an illustration show the reasons why a diagonal preconditioning by Adam can remove axis-aligned ill-conditioning by rescaling and why it cannot do the same if the ill-conditioning is cross coupled. In addition, we present a case study of FINER image fitting architecture that goes over the whole loss landscape analysis framework: the fitting architecture description, reasons due to which its landscape stalls Adam at saddles, the measured PSNR values through our tuned baselines to the $120$--$134$\,dB results of the blockwise second order methods, the error maps behind those numbers, and description of the benefits such image fitting accuracy gives in practice.
A minibatch can influence training beyond the update at which it is observed because AdamW stores past gradient information in its optimizer states. We study this delayed effect through paired trajectories that differ only in one gradient update and share the same subsequent training sequence. We formulate AdamW as a finite-horizon input--state--output (ISO) system whose state contains the model parameters and first- and second-moment estimates. Linearizing the joint dynamics yields a signed response operator that maps a localized gradient perturbation to its future loss effects, revealing how optimizer memory shapes their magnitude, timing, and sign. We further derive an exact multistep error decomposition and establish first-order finite-horizon accuracy under local smoothness and controlled activation switching. Experiments validate the response mechanism and optimizer-state effects, while repeated-future analyses reveal substantial prospective structure in delayed influence that can be partially recovered from ISO approximations. Code is available at https://github.com/Kanyooo/Loss_ISO.
The edge of stability refers to a phenomenon in deep learning with gradient-based optimizers where the Hessian eigenvalues of the loss remain stable above a threshold that the classical descent lemma predicts to be unstable. Previous works formulate the edge of stability with respect to the maximum Hessian eigenvalue and the learning rate. However, we observe that many first-order methods, including gradient descent, significantly violate the stability bound predicted by these theories by a factor as large as $\times 21.1$. Moreover, this deviation turns out to be systematic and highly dependent on the underlying optimizer, which is not captured by previous formulations. This calls for a new formulation of the stability threshold, which we derive from the directional Hessian and the gradient-alignment score with respect to the actual update taken by the optimizer, rather than the maximum curvature mode. Our new formulation of the realized edge of stability not only removes optimizer-dependent offsets and provides more consistent predictions of the stability threshold, but also introduces new diagnostic tools that reveal the unique role of the optimizer in actively balancing between the temporal and spatial budgets in first-order optimization.
Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation. This thesis develops a principled grounding for Adam and motivates new designs. First, we revisit Adam's divergence--convergence debate and show the existence of a problem-dependent phase transition: with properly chosen, batch-size-dependent hyperparameters, Adam converges, whereas under small-$β_2$ regimes it can diverge. Second, we investigate why Adam substantially outperforms SGD on Transformers through Hessian structure. We find that the Hessian evolves toward a near-block-diagonal form along training, accompanied by strong block heterogeneity. We prove that this structure makes Adam's diagonal preconditioner effective. We further show that this special Hessian structure originates from consecutive multiplications of large matrix variables, and we provide a rigorous analysis based on random matrix theory. Finally, these insights motivate Adam-mini, a new optimizer that reduces Adam's memory footprint by 50\% while preserving its performance. Our results also have broader implications beyond Adam: they reveal new local structures in matrix-based nonconvex problems, and also help understand and improve recent NN optimizers, such as Muon.
Ali Janati, Kaoutar El Maghraoui, Andrei Kanavalau +1cs.AI cs.LG
Under the standard split, Muon gets hidden matrices and AdamW embeddings/output head. Muon groks modular addition faster, but its solutions do not hold. All nine configurations on $(a+b) \bmod 113$ grok and later lose generalization. Across five seeds the selected AdamW reference falls below threshold on four, reaching 27.59%. Instability persists across two moduli, two widths, two training fractions, subtraction, and depth. The failure arises at the representation-readout interface, identified only jointly up to an invertible map unselected by the loss. After solving the training set, the gradient falls to order $10^{-6}$ and the optimizers respond differently: step-size elasticity is -0.03 for Muon versus +1.5 for AdamW, and the Muon group moves 8.0 times faster per parameter. From bit-identical states, freezing either group prevents failure. Freezing embeddings/readout removes it in five runs over 451,400 post-grokking steps and five paired seeds: unfrozen arms record 137-321 sub-threshold evaluations, frozen arms none. Removing Muon's normalization and orthogonalization is no substitute: it collapses representation from 326 effective conjugate pairs to 4, shows no recurrent collapse, and fails terminally. Fourier filtering separates circuit failure from masking. Across 43 checkpoints over five seeds and three regimes, the task-aligned family reaches exactly 100% alone. In circuit failure it no longer solves the task; in masking it remains perfect while the full model reaches 45.85%, giving a positive margin on every example, including errors, but being outvoted by a near-equal adversarial remainder. Rescaling it restores 99.9%; grokking is the same condition resolving upward. The task selects the family, swapping $(k,k)$ for $(k,-k)$ under subtraction. Across an abrupt collapse, standard Fourier support is unchanged and the power-distribution cosine remains 0.9899.
For scale-invariant deep networks, Hyperball-style optimizers have shown strong performance in large-scale training by fixing the norms of matrix-valued parameters and normalizing updates. However, the source of their advantage remains unclear. Starting from the angular displacement between consecutive parameter states, we derive an angular effective learning rate that accounts for the parameter-update angle, parameter norm, and update norm. We also show that the conventional norm-based measure is a special case under parameter-update orthogonality. We then decompose optimizer updates into radial and tangential components and analyze how radial updates affect one-step angular displacement. Under the training configurations considered, numerical results show that the radial component has only a limited direct effect on the angular effective learning rate. It therefore cannot explain why MuonH converges more slowly than MuonWD early in training but overtakes it later. To further isolate the underlying mechanism, we devise a heuristic experiment that modifies only the learning-rate schedule so that the dynamics of each optimizer reproduce those of the other. The results suggest that their main difference stems from the evolution of the effective step size rather than an intrinsically superior update direction induced by Hyperball. Our pretraining experiments further show that more aggressive learning-rate decay can accelerate MuonH early in training but may impair its later performance. Thus, maintaining a constant angular velocity does not eliminate the learning-rate-scheduling problem; careful scheduling remains essential to realizing the potential of Hyperball-style optimizers. Our code is publicly available at https://github.com/mangocrazz/hyperball-may-not-be-a-free-lunch.
Interpreting optimizers as gradient-flow discretizations has motivated applying higher-order Runge-Kutta (RK) integrators to neural networks. We build a representative Adam variant (Bogacki-Shampine 3(2) RK pair, FSAL reuse, local-error step control) and evaluate it under a strict compute-matched protocol giving every method the same gradient-evaluation budget - an accounting this literature rarely enforces. Under it the RK variant loses to plain Adam on training loss in both minibatch and full-batch (RK's best-case) training. Instrumenting it shows the "adaptivity" is illusory: normalized error stays far below tolerance, the step size pins at its growth cap from step one (98-100 percent of steps), and no rtol x hmax x h0 setting makes it act; tolerances spanning 100x give bit-identical trajectories. The method is exactly fixed-step Adam with an averaged gradient at 3-4x cost. Repairing it (true reject branch; error on the applied map) reverses the full-batch result - about 40x lower training loss than tuned Adam - and a fixed-step control isolates adaptivity (an emergent warmup-and-growth schedule) as the mechanism. But the gain is fragile to the initial step size and does not reach test accuracy. A pre-registered follow-up rules out the obvious explanations: deeper minimization does not overfit, and an explicit temperature knob only hurts - leaving a trajectory effect, the controller selecting a minimum generalizing 1.3-3.4 points below first-order descent at equal depth. An n=10 study confirms one secondary effect: gradient averaging is a genuine implicit regularizer, beating lr-matched Adam and AdamW on 10/10 seeds - yet RMSprop and NAdam match or beat it at a third the per-step cost. Higher-order adaptive integration buys deeper deterministic minimization and a small regularization effect, but nothing a cheaper, well-tuned first-order baseline does not already provide.
The Muon optimizer reaches the grokking threshold on modular arithmetic faster than AdamW. Prior work attributes this to "spectral-norm constraints plus orthogonalized momentum" but does not isolate which mechanism matters. To better understand Moun's behavior, we run multi-seed and multi-learning-rate sweeps to decompose and stress-test the effect. First, an ablation shows the speedup comes from orthogonalization (the Newton-Schulz iteration): orthogonalize-only matches full Muon, whereas spectral-only is no faster than AdamW and is unreliable, and this verdict holds across learning rates. Second, a mechanistic analysis finds that orthogonalizing optimizers reach generalization at roughly 3x lower spectral norm and, controlling for how much the embedding actually moves, settle into a lower-norm solution rather than simply perturbing the embedding less. Third, reducing the Newton-Schulz iteration count from five to one accelerates reaching the threshold but makes the grokked solution fragile, prone to transient collapse, with fragility that grows with learning rate; a single iteration is fast and stable only at small learning rate, while the canonical five iterations are the learning-rate-robust choice. We also show spectral scaling can be dropped at no measured cost. A methodological thread runs throughout: under a stability-aware metric, "faster" claims about grokking optimizers can invert, so we report both first-crossing and sustained-grok times. To support reproducibility, we release our full training and analysis code at https://github.com/louiswang524/muon-grokking-frontier
Muon has recently emerged as one of the most effective optimizers for training large neural networks, yet its empirical success has been explained from several different perspectives. In this paper, we propose a simple mechanistic interpretation: Muon can be understood as an implicit residual connection during training. Specifically, orthogonalizing the update can sacrifice some immediate gradient fidelity while improving representation preservation for downstream layers. We study this trade-off in controlled linear optimization settings, where Muon can learn representations that are slower to fit a local target but easier for downstream layers to exploit. Our results suggest a conceptual explanation for Muon and a design perspective for optimizers that balance local descent with downstream usability.
Jason R. Brown, Patrick Leask, Lev McKinneycs.LG cs.AI
Emergent misalignment (EM) is a recently discovered phenomenon in LLMs where fine-tuning on a narrow misaligned task, such as writing insecure code, leads to broadly misaligned behaviour on unrelated prompts. Previous work has noted that the severity of EM is highly sensitive to training choices; however, we still lack a systematic characterisation of this sensitivity. We perform a sweep over several Qwen3 models, optimisers, datasets, and batch sizes, and find that the choice of optimiser has the largest effect, producing a 7x spread in misalignment rate. Surprisingly, model size has a negligible effect within the Qwen3 family. An additional sweep over 12 models from three families using Adam confirms that model scale (1B-235B) and family have negligible effects for that optimiser. Analysing the loss-alignment relationship on Qwen3-8B, we find that final log training loss is a strong predictor of alignment, and that stratifying by optimiser captures nearly all the residual variance. Training dynamics reveal that each optimiser follows a different trajectory through loss-alignment space, and that after significant training, the optimiser becomes more important than training loss as a predictor of alignment. Muon, the adaptive optimiser that preserves alignment the best, implicitly regularises for a more uniform distribution of singular values of the LoRA adapter. We evaluate this insight by training with an additional loss term that incentivises a flatter singular value spectrum, and find that this substantially recovers alignment for the more EM-prone adaptive optimisers (Adam and Lion), with negligible cost to training loss. These results identify optimiser choice as a key factor in EM severity, but show that spectral regularisation can substantially mitigate the effects of EM-prone optimisers.
Recently, Muon has gained substantial attention as an appealing alternative to Adam-like optimizers, with many works highlighting its advantages through spectral normalization and improved conditioning. Yet this positive theoretical narrative contrasts with its empirical performance in large language model (LLM) training, where Muon's gains over Adam/AdamW are often mixed, schedule-sensitive, and not uniformly superior. To address this gap, we develop a trajectory-level theory characterizing both the strengths and limitations of Muon. We introduce a mixed-spiked matrix sensing model whose sensing operator decomposes into signal, spike, and bulk components, capturing a mixture of anisotropic structure and long-tail information reminiscent of LLM training. On top of it, we adopted a river-valley perspective in which we view the landscape as composed of a river direction flowing to the desired solution and hill directions encoding nuisance or task-irrelevant information. In the momentum-free setting, we show that Muon moves faster along the information-bearing river direction during early optimization, but can converge much more slowly near the river bottom than gradient descent. We then extend the river-valley perspective to general nonconvex objectives with momentum by studying points on the spectral river. There, while Muon converges faster early on, its orthogonalized update removes residual scale information, making it prone to overshooting and oscillation near the target solution. Together, these results suggest that our characterizations extend beyond spiked matrix sensing and motivate switching to GD-like refinement optimizers in the final phase, rather than relying only on a fixed learning-rate schedule for Muon. We also provide preliminary evidence supporting this two-stage approach in language model training experiments.
Building on a two-parameter Weibull framework for diagnosing transformer weight distributions, we study why the Weibull weight-scale parameter $λ$ grows, overshoots, and then relaxes during AdamW training. We derive a leading-order three-force decomposition of the squared weight norm from the AdamW update: an alignment force measuring the correlation between weights and the adaptive update direction, an injection force from adaptive step magnitude, and a decay force from decoupled weight decay. On self-trained Pythia-70M models with ground-truth optimizer moments, alignment dominates the rise phase, contributing 88-94% of the absolute force budget across four random seeds and remaining robust to super-weight removal. Near saturation, alignment and decay approach balance, explaining the transition from weight-scale growth to relaxation. These force dynamics directly govern the squared-norm component underlying $λ(t)$; the remaining RMS-to-Weibull reconstruction offset is measurable and decomposes into bridge and integration components, totaling approximately 5-6% in densely sampled regions. To extend the analysis to real models where optimizer moments are unavailable, we introduce a spline displacement method that recovers the alignment force from sparse checkpoints with approximately 92-94% accuracy, about twice the naive two-point baseline. We further observe that the peak value of $λ(t)$ varies with training-data coherence in our experiments, suggesting a data-dependent component of weight-scale growth that we leave to a controlled follow-up study. Code and data are available at https://github.com/tiexinding/NPM-Weibull-public.
Shuche Wang, Fengzhuo Zhang, Jiaxiang Li +2cs.LG cs.AI
Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find that Muon and Adam have comparable update norms, so Muon's smaller curvature penalty is driven by lower NDS, not update scale. Third, we study how training data and model structure shape Muon's NDS advantage. Using Zipf-Probabilistic Context-Free Grammar (PCFG) data with controlled imbalance, we show that data imbalance amplifies Muon's NDS advantage over Adam. A within-/cross-layer decomposition further shows that, in the middle and late stages of training, Muon's lower NDS is mainly sustained by smaller within-layer curvature. Beyond empirical evidence, we analyze stylized quadratic problems with heterogeneous curvature and gradient alignment toward high-curvature modes. We prove that Muon attains a smaller average NDS than GD by balancing update energy across curvature groups; when curvature heterogeneity is sufficiently strong, this also yields lower local quadratic loss after the same number of steps.
Muon has recently demonstrated strong empirical performance in large language model training, but the theoretical role of momentum in Muon remains unclear. Existing analyses of Muon either remove momentum to study spectral updates in isolation, or retain momentum without explaining why it improves empirical performance. Our work bridges this gap by showing momentum in Muon acts as a spectral filter. Under a structured signal-plus-perturbation gradient model, we prove that momentum suppresses perturbations while preserving the dominant signal, thereby enlarging the spectral gap between them. This enlarged gap stabilizes the singular subspaces of the matrix passed to Muon's orthogonalization step, making the resulting update more reliable. We further show that applying momentum before orthogonalization achieves provably stronger alignment with the signal component of the gradient than either reversing this order or simply removing momentum. Experiments across diverse tasks, including LLM pretraining, support our theoretical analysis. More broadly, our theory offers a starting point for understanding the benefits of momentum in other matrix-based optimizers.
Gagik Magakyan, Pablo Parrilo, Asuman Ozdaglarcs.LG cs.AI
Orthonormalized update rules have rapidly become a leading choice of optimizer for training large language models, with recent open-source state-of-the-art models adopting Muon. To keep these updates tractable, Muon performs the orthonormalization with the Newton--Schulz (NS) iteration. Since NS is only approximate, directions with small singular values fail to be orthonormalized. In Muon, NS is applied to the momentum matrix at every step, yet little is known about how the singular value spectrum of these momentum matrices behaves during training, or how that behavior changes with model size. We present the first systematic study of this question. Tracking singular value quantiles of the momentum buffer across layers in models ranging from 77M to 2.8B parameters, we observe a consistent picture: after a short burn-in, the quantiles stabilize at a value determined by the layer type and model size. These stabilization values follow remarkably clean power laws in model size, with layer-dependent exponents. Layers up to mid-late depth scale very mildly with model size $M$ (around $M^{-0.25}$), so the standard 5-step NS configuration used at academic scale will continue to orthonormalize them at much larger scales. Some of the late layers, however, scale much more aggressively (up to $M^{-0.96}$) and will fall into the NS failure regime at frontier scale unless one uses more NS iterations or better-tuned coefficients. NS iterations are computationally expensive at scale; our laws give practitioners a principled, layer-aware recipe for choosing the minimum NS configuration that still orthonormalizes the directions that matter -- avoiding unnecessary computation without sacrificing update quality.