Lenka Zdeborová

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https://people.epfl.ch/lenka.zdeborova?lang=fr

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Bayesian reconstruction of memories stored in neural networks from their connectivity

Florent Gérard Krzakala, Lenka Zdeborová

The advent of comprehensive synaptic wiring diagrams of large neural circuits has created the field of connectomics and given rise to a number of open research questions. One such question is whether it is possible to reconstruct the information stored in a recurrent network of neurons, given its synaptic connectivity matrix. Here, we address this question by determining when solving such an inference problem is theoretically possible in specific attractor network models and by providing a practical algorithm to do so. The algorithm builds on ideas from statistical physics to perform approximate Bayesian inference and is amenable to exact analysis. We study its performance on three different models, compare the algorithm to standard algorithms such as PCA, and explore the limitations of reconstructing stored patterns from synaptic connectivity.

2023

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Learning curves for the multi-class teacher-student perceptron

Elisabetta Cornacchia, Bruno Loureiro, Francesca Mignacco, Lenka Zdeborová

One of the most classical results in high-dimensional learning theory provides a closed-form expression for the generalisation error of binary classification with a single-layer teacher-student perceptron on i.i.d. Gaussian inputs. Both Bayes-optimal (BO) estimation and empirical risk minimisation (ERM) were extensively analysed in this setting. At the same time, a considerable part of modern machine learning practice concerns multi-class classification. Yet, an analogous analysis for the multi-class teacher-student perceptron was missing. In this manuscript we fill this gap by deriving and evaluating asymptotic expressions for the BO and ERM generalisation errors in the high-dimensional regime. For Gaussian teacher, we investigate the performance of ERM with both cross-entropy and square losses, and explore the role of ridge regularisation in approaching Bayes-optimality. In particular, we observe that regularised cross-entropy minimisation yields close-to-optimal accuracy. Instead, for Rademacher teacher we show that a first-order phase transition arises in the BO performance.

IOP Publishing Ltd2023

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Uncertainty quantification is a central challenge in reliable and trustworthy machine learning. Naive measures such as last-layer scores are well-known to yield overconfident estimates in the context of overparametrized neural networks. Several methods, ranging from temperature scaling to different Bayesian treatments of neural networks, have been proposed to mitigate overconfidence, most often supported by the numerical observation that they yield better calibrated uncertainty measures. In this work, we provide a sharp comparison between popular uncertainty measures for binary classification in a mathematically tractable model for overparametrized neural networks: the random features model. We discuss a trade-off between classification accuracy and calibration, unveiling a double descent like behavior in the calibration curve of optimally regularized estimators as a function of overparametrization. This is in contrast with the empirical Bayes method, which we show to be well calibrated in our setting despite the higher generalization error and overparametrization.

PMLR Proceedings of Machine Learning Research2023

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