arXiv subjects

P. Rotondo

Publications and source records attributed to P. Rotondo.

7 recordsLinked to original sources

arXiv2024

Kernel shape renormalization explains output-output correlations in finite Bayesian one-hidden-layer networks

Finite-width one hidden layer networks with multiple neurons in the readout layer display non-trivial output-output correlations that vanish in the lazy-training infinite-width limit. In this manuscript we leverage recent progress in the proportional limit of Bayesian deep learning (that is the limit where the size of the training set $P$ and the width of the hidden layers $N$ are taken to infinity keeping their ratio $\alpha = P/N$ finite) to rationalize this empirical evidence. In particular, we show that output-output correlations in finite fully-connected networks are taken into account by a kernel shape renormalization of the infinite-width NNGP kernel, which naturally arises in the proportional limit. We perform accurate numerical experiments both to assess the predictive power of the Bayesian framework in terms of generalization, and to quantify output-output correlations in finite-width networks. By quantitatively matching our predictions with the observed correlations, we provide additional evidence that kernel shape renormalization is instrumental to explain the phenomenology observed in finite Bayesian one hidden layer networks.

cond-mat.dis-nn↗

arXiv2024

Predictive power of a Bayesian effective action for fully-connected one hidden layer neural networks in the proportional limit

We perform accurate numerical experiments with fully-connected (FC) one-hidden layer neural networks trained with a discretized Langevin dynamics on the MNIST and CIFAR10 datasets. Our goal is to empirically determine the regimes of validity of a recently-derived Bayesian effective action for shallow architectures in the proportional limit. We explore the predictive power of the theory as a function of the parameters (the temperature $T$, the magnitude of the Gaussian priors $\lambda_1$, $\lambda_0$, the size of the hidden layer $N_1$ and the size of the training set $P$) by comparing the experimental and predicted generalization error. The very good agreement between the effective theory and the experiments represents an indication that global rescaling of the infinite-width kernel is a main physical mechanism for kernel renormalization in FC Bayesian standard-scaled shallow networks.

cond-mat.dis-nn↗

arXiv2023

Local Kernel Renormalization as a mechanism for feature learning in overparametrized Convolutional Neural Networks

Feature learning, or the ability of deep neural networks to automatically learn relevant features from raw data, underlies their exceptional capability to solve complex tasks. However, feature learning seems to be realized in different ways in fully-connected (FC) or convolutional architectures (CNNs). Empirical evidence shows that FC neural networks in the infinite-width limit eventually outperform their finite-width counterparts. Since the kernel that describes infinite-width networks does not evolve during training, whatever form of feature learning occurs in deep FC architectures is not very helpful in improving generalization. On the other hand, state-of-the-art architectures with convolutional layers achieve optimal performances in the finite-width regime, suggesting that an effective form of feature learning emerges in this case. In this work, we present a simple theoretical framework that provides a rationale for these differences, in one hidden layer networks. First, we show that the generalization performance of a finite-width FC network can be obtained by an infinite-width network, with a suitable choice of the Gaussian priors. Second, we derive a finite-width effective action for an architecture with one convolutional hidden layer and compare it with the result available for FC networks. Remarkably, we identify a completely different form of kernel renormalization: whereas the kernel of the FC architecture is just globally renormalized by a single scalar parameter, the CNN kernel undergoes a local renormalization, meaning that the network can select the local components that will contribute to the final prediction in a data-dependent way. This finding highlights a simple mechanism for feature learning that can take place in overparametrized shallow CNNs, but not in shallow FC architectures or in locally connected neural networks without weight sharing.

cs.LG↗

arXiv2022

A statistical mechanics framework for Bayesian deep neural networks beyond the infinite-width limit

Despite the practical success of deep neural networks, a comprehensive theoretical framework that can predict practically relevant scores, such as the test accuracy, from knowledge of the training data is currently lacking. Huge simplifications arise in the infinite-width limit, where the number of units $N_\ell$ in each hidden layer ($\ell=1,\dots, L$, being $L$ the depth of the network) far exceeds the number $P$ of training examples. This idealisation, however, blatantly departs from the reality of deep learning practice. Here, we use the toolset of statistical mechanics to overcome these limitations and derive an approximate partition function for fully-connected deep neural architectures, which encodes information about the trained models. The computation holds in the ''thermodynamic limit'' where both $N_\ell$ and $P$ are large and their ratio $\alpha_\ell = P/N_\ell$ is finite. This advance allows us to obtain (i) a closed formula for the generalisation error associated to a regression task in a one-hidden layer network with finite $\alpha_1$; (ii) an approximate expression of the partition function for deep architectures (via an ''effective action'' that depends on a finite number of ''order parameters''); (iii) a link between deep neural networks in the proportional asymptotic limit and Student's $t$ processes.

cond-mat.dis-nn↗

arXiv2022

Universal mean field upper bound for the generalisation gap of deep neural networks

Modern deep neural networks (DNNs) represent a formidable challenge for theorists: according to the commonly accepted probabilistic framework that describes their performance, these architectures should overfit due to the huge number of parameters to train, but in practice they do not. Here we employ results from replica mean field theory to compute the generalisation gap of machine learning models with quenched features, in the teacher-student scenario and for regression problems with quadratic loss function. Notably, this framework includes the case of DNNs where the last layer is optimised given a specific realisation of the remaining weights. We show how these results -- combined with ideas from statistical learning theory -- provide a stringent asymptotic upper bound on the generalisation gap of fully trained DNN as a function of the size of the dataset $P$. In particular, in the limit of large $P$ and $N_{\textrm{out}} $ (where $N_\textrm{out}$ is the size of the last layer) and $N_\textrm{out} \ll P$, the generalisation gap approaches zero faster than $2 N_\textrm{out}/P$, for any choice of both architecture and teacher function. Notably, this result greatly improves existing bounds from statistical learning theory. We test our predictions on a broad range of architectures, from toy fully-connected neural networks with few hidden layers to state-of-the-art deep convolutional neural networks.

cond-mat.dis-nn↗

arXiv2018

Singularities in large deviations of work in quantum quenches

We investigate large deviations of the work performed in a quantum quench across two different phases separated by a quantum critical point, using as example the Dicke model quenched from its superradiant to its normal phase. We extract the distribution of the work from the Loschmidt amplitude and compute for both the corresponding large-deviation forms. Comparing these findings with the predictions of the classification scheme put forward in [Phys. Rev. Lett. 109, 250602 (2012)], we are able to identify a regime which is in fact distinct to the ones identified so far: here the rate function exhibits a non-analytical point which is a strong indication of the existence of an out-of-equilibrium phase transition in the rare fluctuations of the work.

cond-mat.stat-mech↗

arXiv2017

Open quantum generalisation of Hopfield neural networks

We propose a new framework to understand how quantum effects may impact on the dynamics of neural networks. We implement the dynamics of neural networks in terms of Markovian open quantum systems, which allows us to treat thermal and quantum coherent effects on the same footing. In particular, we propose an open quantum generalisation of the celebrated Hopfield neural network, the simplest toy model of associative memory. We determine its phase diagram and show that quantum fluctuations give rise to a qualitatively new non-equilibrium phase. This novel phase is characterised by limit cycles corresponding to high-dimensional stationary manifolds that may be regarded as a generalisation of storage patterns to the quantum domain.

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