Alex Massucco

Alex Massucco, Leonardo Del Grande, Marcello Carioni et al.

In recent years, transformer architectures have revolutionized the field of language processing, opening the door to previously unforeseen possibilities. However, from a theoretical point of view, the mathematical models proposed in the literature often lack direct contact with the actual architectures and depend on strong simplifying assumptions. In this paper, we reduce this gap by modelling the data flow in multi-headed transformer architectures as time-dependent gradient flows for a suitable interaction energy capturing the design of the attention mechanism. The explicit dependence on time allows us to consider different weights for each head and for each layer, without imposing constraints on the initialization method. Moreover, we prove that, under a suitable integrability assumption on the evolution of the weights, each element of the $ω$-limit set of the gradient flows is a stationary point of the interaction energy at a limiting weight distribution. Finally, we analyse the stability of the gradient flows considering perturbations of both the initial data and the weights. Specifically, on the one hand, we study the robustness of the proposed models with respect to noisy inputs, establishing a continuous dependence of the gradient flows on the initial data and uniqueness of the flows. On the other hand, we prove the $Γ$-convergence of the perturbed interaction energy to the unperturbed one, leading to the convergence of the corresponding gradient flows. We complement these theoretical results with numerical experiments that confirm the predicted energy-dissipation identity and clarify the asymptotic behavior of the dynamics in both the autonomous-like (Ornstein--Uhlenbeck) and the genuinely non-autonomous (oscillating-weights) regimes.

Zakhar Shumaylov, Nathaël Da Costa, Peter Zaika et al.

The recent empirical success of the Muon optimizer has renewed interest in non-Euclidean optimization, typically justified by similarities with second-order methods, and linear minimization oracle (LMO) theory. In this paper, we challenge this geometric narrative through three contributions, demonstrating that precise geometric structure is not the key factor affecting optimization performance. First, we introduce Freon, a family of optimizers based on Schatten (quasi-)norms, powered by a novel, provably optimal QDWH-based iterative approximation. Freon naturally interpolates between SGD and Muon, while smoothly extrapolating into the quasi-norm regime. Empirically, the best-performing Schatten parameters for GPT-2 lie strictly within the quasi-norm regime, and thus cannot be represented by any unitarily invariant LMO. Second, noting that Freon performs well across a wide range of exponents, we introduce Kaon, an absurd optimizer that replaces singular values with random noise. Despite lacking any coherent geometric structure, Kaon matches Muon's performance and retains classical convergence guarantees, proving that strict adherence to a precise geometry is practically irrelevant. Third, having shown that geometry is not the primary driver of performance, we demonstrate it is instead controlled by two local quantities: alignment and descent potential. Ultimately, each optimizer must tune its step size around these two quantities. While their dynamics are difficult to predict a-priori, evaluating them within a stochastic random feature model yields a precise insight: Muon succeeds not by tracking an ideal global geometry, but by guaranteeing step-size optimality.

Alex Massucco, Davide Murari, Carola-Bibiane Schönlieb

Very deep neural networks achieve state-of-the-art performance by extracting rich, hierarchical features. Yet, training them via backpropagation is often hindered by vanishing or exploding gradients. Existing remedies, such as orthogonal or variance-preserving initialisation and residual architectures, allow for a more stable gradient propagation and the training of deeper models. In this work, we introduce a unified mathematical framework that describes a broad class of nonlinear feedforward and residual networks, whose input-to-output Jacobian matrices are exactly orthogonal almost everywhere. Such a constraint forces the resulting networks to achieve perfect dynamical isometry and train efficiently despite being very deep. Our formulation not only recovers standard architectures as particular cases but also yields new designs that match the trainability of residual networks without relying on conventional skip connections. We provide experimental evidence that perfect Jacobian orthogonality at initialisation is sufficient to stabilise training and achieve competitive performance. We compare this strategy to networks regularised to maintain the Jacobian orthogonality and obtain comparable results. We further extend our analysis to a class of networks well-approximated by those with orthogonal Jacobians and introduce networks with Jacobians representing partial isometries. These generalized models are then showed to maintain the favourable trainability properties.