Stochastic dynamics with random resetting leads to a non-equilibrium steady state. Here, we consider the thermodynamics of resetting by deriving the first and second law for reset processes far from equilibrium. We identify the contributions to the entropy production of the system which arise due to resetting and show that they correspond to the rate with which information is either erased or created. Using Landauer's principle, we derive a bound on the amount of work that is required to maintain a resetting process. We discuss different regimes of resetting, including a Maxwell's demon scenario where heat is extracted from a bath at constant temperature.

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Deep neural networks achieve stellar generalisation even when they have enough parameters to easily fit all their training data. We study this phenomenon by analysing the dynamics and the performance of over-parameterised two-layer neural networks in the teacher–student setup, where one network, the student, is trained on data generated by another network, called the teacher. We show how the dynamics of stochastic gradient descent (SGD) is captured by a set of differential equations and prove that this description is asymptotically exact in the limit of large inputs. Using this framework, we calculate the final generalisation error of student networks that have more parameters than their teachers. We find that the final generalisation error of the student increases with network size when training only the first layer, but stays constant or even decreases with size when training both layers. We show that these different behaviours have their root in the different solutions SGD finds for different activation functions. Our results indicate that achieving good generalisation in neural networks goes beyond the properties of SGD alone and depends on the interplay of at least the algorithm, the model architecture, and the data set.

Teacher-student models provide a powerful framework in which the typical case performance of highdimensional supervised learning tasks can be studied in closed form. In this setting, labels are assigned to data -often taken to be Gaussian i.i.d. -by a teacher model, and the goal is to characterise the typical performance of the student model in recovering the parameters that generated the labels. In this manuscript we discuss a generalisation of this setting where the teacher and student can act on different spaces, generated with fixed, but generic feature maps. This is achieved via the rigorous study of a high-dimensional Gaussian covariate model. Our contribution is two-fold: First, we prove a rigorous formula for the asymptotic training loss and generalisation error achieved by empirical risk minimization for this model. Second, we present a number of situations where the learning curve of the model captures the one of a realistic data set learned with kernel regression and classification, with out-of-the-box feature maps such as random projections or scattering transforms, or with pre-learned ones -such as the features learned by training multi-layer neural networks. We discuss both the power and the limitations of the Gaussian teacher-student framework as a typical case analysis capturing learning curves as encountered in practice on real data sets.

Virtually every organism gathers information about its noisy environment and builds models from those data, mostly using neural networks. Here, we use stochastic thermodynamics to analyze the learning of a classification rule by a neural network. We show that the information acquired by the network is bounded by the thermodynamic cost of learning and introduce a learning efficiency η≤1. We discuss the conditions for optimal learning and analyze Hebbian learning in the thermodynamic limit.

Teacher-student models provide a framework in which the typical-case performance of high-dimensional supervised learning can be described in closed form. The assumptions of Gaussian i.i.d. input data underlying the canonical teacher-student model may, however, be perceived as too restrictive to capture the behaviour of realistic data sets. In this paper, we introduce a Gaussian covariate generalisation of the model where the teacher and student can act on different spaces, generated with fixed, but generic feature maps. While still solvable in a closed form, this generalization is able to capture the learning curves for a broad range of realistic data sets, thus redeeming the potential of the teacher-student framework. Our contribution is then two-fold: first, we prove a rigorous formula for the asymptotic training loss and generalisation error. Second, we present a number of situations where the learning curve of the model captures the one of a realistic data set learned with kernel regression and classification, with out-of-the-box feature maps such as random projections or scattering transforms, or with pre-learned ones—such as the features learned by training multi-layer neural networks. We discuss both the power and the limitations of the framework.

Understanding the impact of data structure on learning in neural networks remains a key challenge for the theory of neural networks. Many theoretical works on neural networks do not explicitly model training data, or assume that inputs are drawn independently from some factorised probability distribution. Here, we go beyond the simple i.i.d. modelling paradigm by studying neural networks trained on data drawn from structured generative models. We make three contributions: First, we establish rigorous conditions under which a class of generative models shares key statistical properties with an appropriately chosen Gaussian feature model. Second, we use this Gaussian equivalence theorem (GET) to derive a closed set of equations that describe the dynamics of twolayer neural networks trained using one-pass stochastic gradient descent on data drawn from a large class of generators. We complement our theoretical results by experiments demonstrating how our theory applies to deep, pre-trained generative models.

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