arXiv subjects

Rosalba Pacelli

Publications and source records attributed to Rosalba Pacelli.

5 recordsLinked to original sources

arXiv2026

Kernel Renormalization in Bayesian Deep Neural Networks: the Equivalent Wishart Ansatz in the Proportional Regime

The scaling limit where both the size of the training set $P$ and the width $N$ of a deep neural network grow at the same rate, the so-called proportional-width regime, has been intensely studied for shallow, single-hidden-layer networks. However, extending these non-perturbative results from shallow architectures to deep non-linear networks has proven very challenging. Here we present an effective approximate approach to predict the generalization performance of Bayesian multi-layer perceptrons (MLPs) of fixed depth $L$ on arbitrary high-dimensional data. We propose an equivalent Wishart Ansatz to capture the dominant stochastic fluctuations of the hierarchical empirical kernels of MLPs. This allows us to perform a large deviation analysis for the partition function of MLPs in the proportional limit, expressed in terms of a renormalized NNGP kernel. In this description, even strong representation learning in the proportional limit is encoded in at most $L$ scalar order parameters, determined self-consistently. Extending the approach to convolutional architectures (CNNs), we identify a hierarchical local kernel renormalization mechanism, which allows to quantify more complex data-dependent transformations of the large-width kernel in CNNs due to finite-width effects. We test our effective theory against sampling experiments from the Bayesian posterior of finite deep neural networks with depths $L \sim O(10)$ and $P\sim O(10^3)$ on classic benchmark datasets, finding overall very good agreement together with two distinct types of systematic deviations.

cs.LG↗

arXiv2025

Microscopic and collective signatures of feature learning in neural networks

Feature extraction - the ability to identify relevant properties of data - is a key factor underlying the success of deep learning. Yet, it has proved difficult to elucidate its nature within existing predictive theories, to the extent that there is no consensus on the very definition of feature learning. A promising hint in this direction comes from previous phenomenological observations of quasi-universal aspects in the training dynamics of neural networks, displayed by simple properties of feature geometry. We address this problem within a statistical-mechanics framework for Bayesian learning in one hidden layer neural networks with standard parameterization. Analytical computations in the proportional limit (when both the network width and the size of the training set are large) can quantify fingerprints of feature learning, both collective ones (related to manifold geometry) and microscopic ones (related to the weights). In particular, (i) the distance between different class manifolds in feature space is a nonmonotonic function of the temperature, which we interpret as the equilibrium counterpart of a phenomenon observed under gradient descent (GD) dynamics, and (ii) the microscopic learnable parameters in the network undergo a finite data-dependent displacement with respect to the infinite-width limit, and develop correlations. These results indicate that nontrivial feature learning is at play in a regime where the posterior predictive distribution is that of Gaussian process regression with a trivially rescaled prior.

cond-mat.dis-nn↗

arXiv2024

Statistical mechanics of transfer learning in fully-connected networks in the proportional limit

Transfer learning (TL) is a well-established machine learning technique to boost the generalization performance on a specific (target) task using information gained from a related (source) task, and it crucially depends on the ability of a network to learn useful features. Leveraging recent analytical progress in the proportional regime of deep learning theory (i.e. the limit where the size of the training set $P$ and the size of the hidden layers $N$ are taken to infinity keeping their ratio $α= P/N$ finite), in this work we develop a novel single-instance Franz-Parisi formalism that yields an effective theory for TL in fully-connected neural networks. Unlike the (lazy-training) infinite-width limit, where TL is ineffective, we demonstrate that in the proportional limit TL occurs due to a renormalized source-target kernel that quantifies their relatedness and determines whether TL is beneficial for generalization.

cond-mat.dis-nn↗

arXiv2024

Les Houches Lectures on Deep Learning at Large & Infinite Width

These lectures, presented at the 2022 Les Houches Summer School on Statistical Physics and Machine Learning, focus on the infinite-width limit and large-width regime of deep neural networks. Topics covered include various statistical and dynamical properties of these networks. In particular, the lecturers discuss properties of random deep neural networks; connections between trained deep neural networks, linear models, kernels, and Gaussian processes that arise in the infinite-width limit; and perturbative and non-perturbative treatments of large but finite-width networks, at initialization and after training.

stat.ML↗

arXiv2022

Learning through atypical "phase transitions" in overparameterized neural networks

Current deep neural networks are highly overparameterized (up to billions of connection weights) and nonlinear. Yet they can fit data almost perfectly through variants of gradient descent algorithms and achieve unexpected levels of prediction accuracy without overfitting. These are formidable results that defy predictions of statistical learning and pose conceptual challenges for non-convex optimization. In this paper, we use methods from statistical physics of disordered systems to analytically study the computational fallout of overparameterization in non-convex binary neural network models, trained on data generated from a structurally simpler but "hidden" network. As the number of connection weights increases, we follow the changes of the geometrical structure of different minima of the error loss function and relate them to learning and generalization performance. A first transition happens at the so-called interpolation point, when solutions begin to exist (perfect fitting becomes possible). This transition reflects the properties of typical solutions, which however are in sharp minima and hard to sample. After a gap, a second transition occurs, with the discontinuous appearance of a different kind of "atypical" structures: wide regions of the weight space that are particularly solution-dense and have good generalization properties. The two kinds of solutions coexist, with the typical ones being exponentially more numerous, but empirically we find that efficient algorithms sample the atypical, rare ones. This suggests that the atypical phase transition is the relevant one for learning. The results of numerical tests with realistic networks on observables suggested by the theory are consistent with this scenario.

cs.LG↗