Alessandro Breccia

Alessandro Breccia, Federica Gerace, Marco Lippi et al.

We study whether a Large Language Model can learn the deterministic sequence of trees generated by the iterated prime factorization of the natural numbers. Each integer is mapped into a rooted planar tree and the resulting sequence $ \mathbb{N}\mathcal{T}$ defines an arithmetic text with measurable statistical structure. A transformer network (the GPT-2 architecture) is trained from scratch on the first $10^{11}$ elements to subsequently test its predictive ability under next-word and masked-word prediction tasks. Our results show that the model partially learns the internal grammar of $\mathbb{N}\mathcal{T}$, capturing non-trivial regularities and correlations. This suggests that learnability may extend beyond empirical data to the very structure of arithmetic.

Leena Chennuru Vankadara, Moritz Haas, Luke Hayward et al.

Recent frontier large language models predominantly rely on Mixture-of-Experts (MoE) architectures. Despite empirical progress, there is still no principled understanding of how hyperparameters should scale with network width $N$, expert width $N_e$, number of experts $M$, sparsity $K$, and depth $L$ to ensure both stability and optimal performance at scale. We take a principled step toward resolving this gap by analyzing three different scaling regimes: (I) co-scaling $N\asymp N_e$, (II) co-scaling $N\asymp M\asymp K$, and (III) full proportional scaling of $N, N_e, M$, and $K$. For each regime, we develop a novel Dynamical Mean Field Theory (DMFT) description of the limiting training dynamics of MoEs that provides a formal foundation for our analysis. Within this framework, we derive the unique parameterization for SGD and Adam satisfying all maximal-update ($μ$) desiderata. We then show that the resulting $μ$P prescription does not reliably induce monotonic improvement with scale or robust learning-rate transfer. We trace these pathologies to scale-dependent observables in the aggregation dynamics, which motivates a refined set of desiderata that we term maximal scale stability. Guided by this principle, we derive a Maximally Scale-Stable Parameterization (MSSP) for both SGD and Adam in all three scaling regimes, and characterize the corresponding limiting dynamics - qualitatively distinct from the $μ$P limit - through a separate DMFT analysis. Experiments verify that MSSP robustly recovers learning rate transfer and monotonic improvement with scale across regimes. Combined with existing depth-scaling theory, these results provide a complete scaling prescription for MoE architectures as a function of width, depth, expert width, and number of experts.

Jacopo Graldi, Alessandro Breccia, Giulia Lanzillotta et al.

Despite recent efforts, neural networks still struggle to learn in non-stationary environments, and our understanding of catastrophic forgetting (CF) is far from complete. In this work, we perform a systematic study on the impact of model scale and the degree of feature learning in continual learning. We reconcile existing contradictory observations on scale in the literature, by differentiating between lazy and rich training regimes through a variable parameterization of the architecture. We show that increasing model width is only beneficial when it reduces the amount of feature learning, yielding more laziness. Using the framework of dynamical mean field theory, we then study the infinite width dynamics of the model in the feature learning regime and characterize CF, extending prior theoretical results limited to the lazy regime. We study the intricate relationship between feature learning, task non-stationarity, and forgetting, finding that high feature learning is only beneficial with highly similar tasks. We identify a transition modulated by task similarity where the model exits an effectively lazy regime with low forgetting to enter a rich regime with significant forgetting. Finally, our findings reveal that neural networks achieve optimal performance at a critical level of feature learning, which depends on task non-stationarity and transfers across model scales. This work provides a unified perspective on the role of scale and feature learning in continual learning.

Enrico Lupi, Abhishek, Max Aehle et al.

We simulate hadrons impinging on a homogeneous lead-tungstate (PbWO4) calorimeter to investigate how the resulting light yield and its temporal structure, as detected by an array of light-sensitive sensors, can be processed by a neuromorphic computing system. Our model encodes temporal photon distributions as spike trains and employs a fully connected spiking neural network to estimate the total deposited energy, as well as the position and spatial distribution of the light emissions within the sensitive material. The extracted primitives offer valuable topological information about the shower development in the material, achieved without requiring a segmentation of the active medium. A potential nanophotonic implementation using III-V semiconductor nanowires is discussed. It can be both fast and energy efficient.