Alessandro Favero

Antonio Sclocchi, Alessandro Favero, Matthieu Wyart · cambridge

Understanding the structure of real data is paramount in advancing modern deep-learning methodologies. Natural data such as images are believed to be composed of features organized in a hierarchical and combinatorial manner, which neural networks capture during learning. Recent advancements show that diffusion models can generate high-quality images, hinting at their ability to capture this underlying compositional structure. We study this phenomenon in a hierarchical generative model of data. We find that the backward diffusion process acting after a time $t$ is governed by a phase transition at some threshold time, where the probability of reconstructing high-level features, like the class of an image, suddenly drops. Instead, the reconstruction of low-level features, such as specific details of an image, evolves smoothly across the whole diffusion process. This result implies that at times beyond the transition, the class has changed, but the generated sample may still be composed of low-level elements of the initial image. We validate these theoretical insights through numerical experiments on class-unconditional ImageNet diffusion models. Our analysis characterizes the relationship between time and scale in diffusion models and puts forward generative models as powerful tools to model combinatorial data properties.

Alessandro Favero, Antonio Sclocchi, Francesco Cagnetta et al. · cambridge

Natural data is often organized as a hierarchical composition of features. How many samples do generative models need in order to learn the composition rules, so as to produce a combinatorially large number of novel data? What signal in the data is exploited to learn those rules? We investigate these questions in the context of diffusion models both theoretically and empirically. Theoretically, we consider a simple probabilistic context-free grammar - a tree-like graphical model used to represent the hierarchical and compositional structure of data such as language and images. We demonstrate that diffusion models learn the grammar's composition rules with the sample complexity required for clustering features with statistically similar context, a process similar to the word2vec algorithm. However, this clustering emerges hierarchically: higher-level features associated with longer contexts require more data to be identified. This mechanism leads to a sample complexity that scales polynomially with the said context size. As a result, diffusion models trained on an intermediate dataset size generate data coherent up to a certain scale, but lacking global coherence. We test these predictions across different domains and find remarkable agreement: both generated texts and images achieve progressively larger coherence lengths as the training time or dataset size grows. We discuss connections between the hierarchical clustering mechanism we introduce here and the renormalization group in physics.

Antonio Sclocchi, Alessandro Favero, Noam Itzhak Levi et al. · cambridge

High-dimensional data must be highly structured to be learnable. Although the compositional and hierarchical nature of data is often put forward to explain learnability, quantitative measurements establishing these properties are scarce. Likewise, accessing the latent variables underlying such a data structure remains a challenge. In this work, we show that forward-backward experiments in diffusion-based models, where data is noised and then denoised to generate new samples, are a promising tool to probe the latent structure of data. We predict in simple hierarchical models that, in this process, changes in data occur by correlated chunks, with a length scale that diverges at a noise level where a phase transition is known to take place. Remarkably, we confirm this prediction in both text and image datasets using state-of-the-art diffusion models. Our results show how latent variable changes manifest in the data and establish how to measure these effects in real data using diffusion models.

Amel Abdelraheem, Alessandro Favero, Gerome Bovet et al. · cambridge

Foundation models have revolutionized computer vision by enabling broad generalization across diverse tasks. Yet, they remain highly susceptible to adversarial perturbations and targeted backdoor attacks. Mitigating such vulnerabilities remains an open challenge, especially given that the large-scale nature of the models prohibits retraining to ensure safety. Existing backdoor removal approaches rely on costly fine-tuning to override the harmful behavior, and can often degrade performance on other unrelated tasks. This raises the question of whether backdoors can be removed without compromising the general capabilities of the models. In this work, we address this question and study how backdoors are encoded in the model weight space, finding that they are disentangled from other benign tasks. Specifically, this separation enables the isolation and erasure of the backdoor's influence on the model with minimal impact on clean performance. Building on this insight, we introduce a simple unlearning method that leverages such disentanglement. Through extensive experiments with CLIP-based models and common adversarial triggers, we show that, given the knowledge of the attack, our method achieves approximately perfect unlearning, while retaining, on average, 96% of clean accuracy. Additionally, we demonstrate that even when the attack and its presence are unknown, our method successfully unlearns backdoors by proper estimation using reverse-engineered triggers. Overall, our method consistently yields better unlearning and clean accuracy tradeoffs when compared to present state-of-the-art defenses.

Alessandro Favero · cambridge

Deep neural networks have achieved remarkable success, yet our understanding of how they learn remains limited. These models can learn high-dimensional tasks, which is generally statistically intractable due to the curse of dimensionality. This apparent paradox suggests that learnable data must have an underlying latent structure. What is the nature of this structure? How do neural networks encode and exploit it, and how does it quantitatively impact performance - for instance, how does generalization improve with the number of training examples? This thesis addresses these questions by studying the roles of locality and compositionality in data, tasks, and deep learning representations.

Francesco Cagnetta, Alessandro Favero, Antonio Sclocchi et al. · cambridge

How do neural language models acquire a language's structure when trained for next-token prediction? We address this question by deriving theoretical scaling laws for neural network performance on synthetic datasets generated by the Random Hierarchy Model (RHM) -- an ensemble of probabilistic context-free grammars designed to capture the hierarchical structure of natural language while remaining analytically tractable. Previously, we developed a theory of representation learning based on data correlations that explains how deep learning models capture the hierarchical structure of the data sequentially, one layer at a time. Here, we extend our theoretical framework to account for architectural differences. In particular, we predict and empirically validate that convolutional networks, whose structure aligns with that of the generative process through locality and weight sharing, enjoy a faster scaling of performance compared to transformer models, which rely on global self-attention mechanisms. This finding clarifies the architectural biases underlying neural scaling laws and highlights how representation learning is shaped by the interaction between model architecture and the statistical properties of data.

Alessandro Favero, Francesco Cagnetta, Matthieu Wyart

Convolutional neural networks perform a local and translationally-invariant treatment of the data: quantifying which of these two aspects is central to their success remains a challenge. We study this problem within a teacher-student framework for kernel regression, using `convolutional' kernels inspired by the neural tangent kernel of simple convolutional architectures of given filter size. Using heuristic methods from physics, we find in the ridgeless case that locality is key in determining the learning curve exponent $β$ (that relates the test error $ε_t\sim P^{-β}$ to the size of the training set $P$), whereas translational invariance is not. In particular, if the filter size of the teacher $t$ is smaller than that of the student $s$, $β$ is a function of $s$ only and does not depend on the input dimension. We confirm our predictions on $β$ empirically. We conclude by proving, using a natural universality assumption, that performing kernel regression with a ridge that decreases with the size of the training set leads to similar learning curve exponents to those we obtain in the ridgeless case.

Leonardo Petrini, Alessandro Favero, Mario Geiger et al.

Understanding why deep nets can classify data in large dimensions remains a challenge. It has been proposed that they do so by becoming stable to diffeomorphisms, yet existing empirical measurements support that it is often not the case. We revisit this question by defining a maximum-entropy distribution on diffeomorphisms, that allows to study typical diffeomorphisms of a given norm. We confirm that stability toward diffeomorphisms does not strongly correlate to performance on benchmark data sets of images. By contrast, we find that the stability toward diffeomorphisms relative to that of generic transformations $R_f$ correlates remarkably with the test error $ε_t$. It is of order unity at initialization but decreases by several decades during training for state-of-the-art architectures. For CIFAR10 and 15 known architectures, we find $ε_t\approx 0.2\sqrt{R_f}$, suggesting that obtaining a small $R_f$ is important to achieve good performance. We study how $R_f$ depends on the size of the training set and compare it to a simple model of invariant learning.