Leonardo Del Grande

Leonardo Del Grande, Christoph Brune, Marcello Carioni

In this paper, we study total variation (TV)-regularized training of infinite-width shallow ReLU neural networks, formulated as a convex optimization problem over measures on the unit sphere. Our approach leverages the duality theory of TV-regularized optimization problems to establish rigorous guarantees on the sparsity of the solutions to the training problem. Our analysis further characterizes how and when this sparsity persists in a low noise regime and for small regularization parameter. The key observation that motivates our analysis is that, for ReLU activations, the associated dual certificate is piecewise linear in the weight space. Its linearity regions, which we name dual regions, are determined by the activation patterns of the data via the induced hyperplane arrangement. Taking advantage of this structure, we prove that, on each dual region, the dual certificate admits at most one extreme value. As a consequence, the support of any minimizer is finite, and its cardinality can be bounded from above by a constant depending only on the geometry of the data-induced hyperplane arrangement. Then, we further investigate sufficient conditions ensuring uniqueness of such sparse solution. Finally, under a suitable non-degeneracy condition on the dual certificate along the boundaries of the dual regions, we prove that in the presence of low label noise and for small regularization parameter, solutions to the training problem remain sparse with the same number of Dirac deltas. Additionally, their location and the amplitudes converge, and, in case the locations lie in the interior of a dual region, the convergence happens with a rate that depends linearly on the noise and the regularization parameter.

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.