Different gradient-based methods for optimizing overparameterized models can all achieve zero training error yet converge to distinctly different solutions inducing different generalization properties. We provide the first complete characterization of implicit optimization bias for p-norm normalized steepest descent (NSD) and momentum steepest descent (NMD) algorithms in multi-class linear classification with cross-entropy loss. Our key theoretical contribution is proving that these algorithms converge to solutions maximizing the margin with respect to the classifier matrix's p-norm, with established convergence rates. These results encompass important special cases including Spectral Descent and Muon, which we show converge to max-margin solutions with respect to the spectral norm. A key insight of our contribution is that the analysis of general entry-wise and Schatten p-norms can be reduced to the analysis of NSD/NMD with max-norm by exploiting a natural ordering property between all p-norms relative to the max-norm and its dual sum-norm. For the specific case of descent with respect to the max-norm, we further extend our analysis to include preconditioning, showing that Adam converges to the matrix's max-norm solution. Our results demonstrate that the multi-class linear setting, which is inherently richer than the binary counterpart, provides the most transparent framework for studying implicit biases of matrix-parameter optimization algorithms.

We study the implicit bias of momentum-based optimizers on homogeneous models. We first extend existing results on the implicit bias of steepest descent in homogeneous models to normalized steepest descent with an optional learning rate schedule. We then show that for smooth homogeneous models, momentum steepest descent algorithms like Muon (spectral norm), MomentumGD ($\ell_2$ norm), and Signum ($\ell_\infty$ norm) are approximate steepest descent trajectories under a decaying learning rate schedule, proving that these algorithms too have a bias towards KKT points of the corresponding margin maximization problem. We extend the analysis to Adam (without the stability constant), which maximizes the $\ell_\infty$ margin, and to Muon-Signum and Muon-Adam, which maximize a hybrid norm. Our experiments corroborate the theory and show that the identity of the margin maximized depends on the choice of optimizer. Overall, our results extend earlier lines of work on steepest descent in homogeneous models and momentum-based optimizers in linear models.

To define a steepest descent method over a neural network, we need to choose a norm for each layer, a way to aggregate these norms across layers, and whether to use normalization. We systematically explore different alternatives for aggregating norms across layers, both formalizing existing combinations of Adam and the recently proposed Muon as a type of non-Euclidean gradient descent, and deriving new variants of the Muon optimizer. Through a comprehensive experimental evaluation of the optimizers within our framework, we find that Muon is sensitive to the choice of learning rate, whereas a new variant we call MuonMax is significantly more robust. We then show how to combine any non-Euclidean gradient method with model based momentum (known as Momo). The new Momo variants of Muon are significantly more robust to hyperparameter tuning, and often achieve a better validation score. Thus for new tasks, where the optimal hyperparameters are not known, we advocate for using Momo in combination with MuonMax to save on costly hyperparameter tuning.

Training - the optimisation of complex models - is traditionally performed through small, local, iterative updates [D. E. Rumelhart, G. E. Hinton, R. J. Williams, Nature 323, 533-536 (1986)]. Approximating solutions through truncated gradients is a paradigm dating back to Cauchy [A.-L. Cauchy, Comptes Rendus Mathématique 25, 536-538 (1847)] and Newton [I. Newton, The Method of Fluxions and Infinite Series (Henry Woodfall, London, 1736)]. This work introduces the Method of Infinite Descent, a semi-analytic optimisation paradigm that reformulates training as the direct solution to the first-order optimality condition. By analytical resummation of its Taylor expansion, this method yields an exact, algebraic equation for the update step. Realisation of the infinite Taylor tower's cascading resummation is formally derived, and an exploitative algorithm for the direct solve step is proposed. This principle is demonstrated with the herein-introduced AION (Analytic, Infinitely-Optimisable Network) architecture. AION is a model designed expressly to satisfy the algebraic closure required by Infinite Descent. In a simple test problem, AION reaches the optimum in a single descent step. Together, this optimiser-model pair exemplify how analytic structure enables exact, non-iterative convergence. Infinite Descent extends beyond this example, applying to any appropriately closed architecture. This suggests a new class of semi-analytically optimisable models: the \emph{Infinity Class}; sufficient conditions for class membership are discussed. This offers a pathway toward non-iterative learning.

(Lemma 1). (Theorem 1). Reviewer 1 Thank you for your valuable comments. Table 1 was a surprising empirical observation. On dividing out the gradient scale --this approach (taken by Adam) requires more learning rate tuning than Fromage.

Deep learning optimizers are often motivated through a mix of convex and approximate second-order theory. We select three such methods -- Adam, Shampoo and Prodigy -- and argue that each method can instead be understood as a squarely first-order method without convexity assumptions. In fact, after switching off exponential moving averages, each method is equivalent to steepest descent under a particular norm. By generalizing this observation, we chart a new design space for training algorithms. Different operator norms should be assigned to different tensors based on the role that the tensor plays within the network. For example, while linear and embedding layers may have the same weight space of $\mathbb{R}^{m\times n}$, these layers play different roles and should be assigned different norms. We hope that this idea of carefully metrizing the neural architecture might lead to more stable, scalable and indeed faster training.

We propose a new accelerated first-order method for convex optimization under non-Euclidean smoothness assumptions. In contrast to standard acceleration techniques, our approach uses primal-dual iterate sequences taken with respect to differing norms, which are then coupled using an implicitly determined interpolation parameter. For $\ell_p$ norm smooth problems in $d$ dimensions, our method provides an iteration complexity improvement of up to $O(d^{1-\frac{2}{p}})$ in terms of calls to a first-order oracle, thereby allowing us to circumvent long-standing barriers in accelerated non-Euclidean steepest descent.

This paper shows that a wide class of effective learning rules -- those that improve a scalar performance measure over a given time window -- can be rewritten as natural gradient descent with respect to a suitably defined loss function and metric. Specifically, we show that parameter updates within this class of learning rules can be expressed as the product of a symmetric positive definite matrix (i.e., a metric) and the negative gradient of a loss function. We also demonstrate that these metrics have a canonical form and identify several optimal ones, including the metric that achieves the minimum possible condition number. The proofs of the main results are straightforward, relying only on elementary linear algebra and calculus, and are applicable to continuous-time, discrete-time, stochastic, and higher-order learning rules, as well as loss functions that explicitly depend on time.

We study the implicit bias of optimization in robust empirical risk minimization (robust ERM) and its connection with robust generalization. In classification settings under adversarial perturbations with linear models, we study what type of regularization should ideally be applied for a given perturbation set to improve (robust) generalization. We then show that the implicit bias of optimization in robust ERM can significantly affect the robustness of the model and identify two ways this can happen; either through the optimization algorithm or the architecture. We verify our predictions in simulations with synthetic data and experimentally study the importance of implicit bias in robust ERM with deep neural networks.