arXiv subjects

S. Ariosto

Publications and source records attributed to S. Ariosto.

2 recordsLinked to original sources

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↗