Samuele Saviozzi

Andrea Agazzi, Giuseppe Bruno, Eloy Mosig García et al.

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.

Andrea Agazzi, Vittorio Carlei, Marco Romito et al.

Global optimization, particularly for non-convex functions with multiple local minima, poses significant challenges for traditional gradient-based methods. While metaheuristic approaches offer empirical effectiveness, they often lack theoretical convergence guarantees and may disregard available gradient information. This paper introduces a novel gradient-based swarm particle optimization method designed to efficiently escape local minima and locate global optima. Our approach leverages a "Soft-min Energy" interacting function, $J_β(\mathbf{x})$, which provides a smooth, differentiable approximation of the minimum function value within a particle swarm. We define a stochastic gradient flow in the particle space, incorporating a Brownian motion term for exploration and a time-dependent parameter $β$ to control smoothness, similar to temperature annealing. We theoretically demonstrate that for strongly convex functions, our dynamics converges to a stationary point where at least one particle reaches the global minimum, with other particles exhibiting exploratory behavior. Furthermore, we show that our method facilitates faster transitions between local minima by reducing effective potential barriers with respect to Simulated Annealing. More specifically, we estimate the hitting times of unexplored potential wells for our model in the small noise regime and show that they compare favorably with the ones of overdamped Langevin. Numerical experiments on benchmark functions, including double wells and the Ackley function, validate our theoretical findings and demonstrate better performance over the well-known Simulated Annealing method in terms of escaping local minima and achieving faster convergence.