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A theory of data variability in Neural Network Bayesian inference
Jul 31, 2023
Bayesian inference and kernel methods are well established in machine learning. The neural network Gaussian process in particular provides a concept to investigate neural networks in the limit of infinitely wide hidden layers by using kernel and inference methods. Here we build upon this limit and provide a field-theoretic formalism which covers the generalization properties of infinitely wide networks. We systematically compute generalization properties of linear, non-linear, and deep non-linear networks for kernel matrices with heterogeneous entries. In contrast to currently employed spectral methods we derive the generalization properties from the statistical properties of the input, elucidating the interplay of input dimensionality, size of the training data set, and variability of the data. We show that data variability leads to a non-Gaussian action reminiscent of a ($\varphi^3+\varphi^4$)-theory. Using our formalism on a synthetic task and on MNIST we obtain a homogeneous kernel matrix approximation for the learning curve as well as corrections due to data variability which allow the estimation of the generalization properties and exact results for the bounds of the learning curves in the case of infinitely many training data points.
Speed Limits for Deep Learning
Jul 27, 2023
State-of-the-art neural networks require extreme computational power to train. It is therefore natural to wonder whether they are optimally trained. Here we apply a recent advancement in stochastic thermodynamics which allows bounding the speed at which one can go from the initial weight distribution to the final distribution of the fully trained network, based on the ratio of their Wasserstein-2 distance and the entropy production rate of the dynamical process connecting them. Considering both gradient-flow and Langevin training dynamics, we provide analytical expressions for these speed limits for linear and linearizable neural networks e.g. Neural Tangent Kernel (NTK). Remarkably, given some plausible scaling assumptions on the NTK spectra and spectral decomposition of the labels -- learning is optimal in a scaling sense. Our results are consistent with small-scale experiments with Convolutional Neural Networks (CNNs) and Fully Connected Neural networks (FCNs) on CIFAR-10, showing a short highly non-optimal regime followed by a longer optimal regime.
Optimal signal propagation in ResNets through residual scaling
May 12, 2023
Residual networks (ResNets) have significantly better trainability and thus performance than feed-forward networks at large depth. Introducing skip connections facilitates signal propagation to deeper layers. In addition, previous works found that adding a scaling parameter for the residual branch further improves generalization performance. While they empirically identified a particularly beneficial range of values for this scaling parameter, the associated performance improvement and its universality across network hyperparameters yet need to be understood. For feed-forward networks (FFNets), finite-size theories have led to important insights with regard to signal propagation and hyperparameter tuning. We here derive a systematic finite-size theory for ResNets to study signal propagation and its dependence on the scaling for the residual branch. We derive analytical expressions for the response function, a measure for the network's sensitivity to inputs, and show that for deep networks the empirically found values for the scaling parameter lie within the range of maximal sensitivity. Furthermore, we obtain an analytical expression for the optimal scaling parameter that depends only weakly on other network hyperparameters, such as the weight variance, thereby explaining its universality across hyperparameters. Overall, this work provides a framework for theory-guided optimal scaling in ResNets and, more generally, provides the theoretical framework to study ResNets at finite widths.
Neuronal architecture extracts statistical temporal patterns
Jan 24, 2023
Neuronal systems need to process temporal signals. We here show how higher-order temporal (co-)fluctuations can be employed to represent and process information. Concretely, we demonstrate that a simple biologically inspired feedforward neuronal model is able to extract information from up to the third order cumulant to perform time series classification. This model relies on a weighted linear summation of synaptic inputs followed by a nonlinear gain function. Training both - the synaptic weights and the nonlinear gain function - exposes how the non-linearity allows for the transfer of higher order correlations to the mean, which in turn enables the synergistic use of information encoded in multiple cumulants to maximize the classification accuracy. The approach is demonstrated both on a synthetic and on real world datasets of multivariate time series. Moreover, we show that the biologically inspired architecture makes better use of the number of trainable parameters as compared to a classical machine-learning scheme. Our findings emphasize the benefit of biological neuronal architectures, paired with dedicated learning algorithms, for the processing of information embedded in higher-order statistical cumulants of temporal (co-)fluctuations.
Origami in N dimensions: How feed-forward networks manufacture linear separability
Mar 21, 2022
Neural networks can implement arbitrary functions. But, mechanistically, what are the tools at their disposal to construct the target? For classification tasks, the network must transform the data classes into a linearly separable representation in the final hidden layer. We show that a feed-forward architecture has one primary tool at hand to achieve this separability: progressive folding of the data manifold in unoccupied higher dimensions. The operation of folding provides a useful intuition in low-dimensions that generalizes to high ones. We argue that an alternative method based on shear, requiring very deep architectures, plays only a small role in real-world networks. The folding operation, however, is powerful as long as layers are wider than the data dimensionality, allowing efficient solutions by providing access to arbitrary regions in the distribution, such as data points of one class forming islands within the other classes. We argue that a link exists between the universal approximation property in ReLU networks and the fold-and-cut theorem (Demaine et al., 1998) dealing with physical paper folding. Based on the mechanistic insight, we predict that the progressive generation of separability is necessarily accompanied by neurons showing mixed selectivity and bimodal tuning curves. This is validated in a network trained on the poker hand task, showing the emergence of bimodal tuning curves during training. We hope that our intuitive picture of the data transformation in deep networks can help to provide interpretability, and discuss possible applications to the theory of convolutional networks, loss landscapes, and generalization. TL;DR: Shows that the internal processing of deep networks can be thought of as literal folding operations on the data distribution in the N-dimensional activation space. A link to a well-known theorem in origami theory is provided.
Decomposing neural networks as mappings of correlation functions
Feb 10, 2022
Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus non-random weights. To address this issue, we study the mapping between probability distributions implemented by a deep feed-forward network. We characterize this mapping as an iterated transformation of distributions, where the non-linearity in each layer transfers information between different orders of correlation functions. This allows us to identify essential statistics in the data, as well as different information representations that can be used by neural networks. Applied to an XOR task and to MNIST, we show that correlations up to second order predominantly capture the information processing in the internal layers, while the input layer also extracts higher-order correlations from the data. This analysis provides a quantitative and explainable perspective on classification.
Unified Field Theory for Deep and Recurrent Neural Networks
Jan 07, 2022
Understanding capabilities and limitations of different network architectures is of fundamental importance to machine learning. Bayesian inference on Gaussian processes has proven to be a viable approach for studying recurrent and deep networks in the limit of infinite layer width, $n\to\infty$. Here we present a unified and systematic derivation of the mean-field theory for both architectures that starts from first principles by employing established methods from statistical physics of disordered systems. The theory elucidates that while the mean-field equations are different with regard to their temporal structure, they yet yield identical Gaussian kernels when readouts are taken at a single time point or layer, respectively. Bayesian inference applied to classification then predicts identical performance and capabilities for the two architectures. Numerically, we find that convergence towards the mean-field theory is typically slower for recurrent networks than for deep networks and the convergence speed depends non-trivially on the parameters of the weight prior as well as the depth or number of time steps, respectively. Our method exposes that Gaussian processes are but the lowest order of a systematic expansion in $1/n$. The formalism thus paves the way to investigate the fundamental differences between recurrent and deep architectures at finite widths $n$.
Unfolding recurrence by Green's functions for optimized reservoir computing
Oct 14, 2020
Cortical networks are strongly recurrent, and neurons have intrinsic temporal dynamics. This sets them apart from deep feed-forward networks. Despite the tremendous progress in the application of feed-forward networks and their theoretical understanding, it remains unclear how the interplay of recurrence and non-linearities in recurrent cortical networks contributes to their function. The purpose of this work is to present a solvable recurrent network model that links to feed forward networks. By perturbative methods we transform the time-continuous, recurrent dynamics into an effective feed-forward structure of linear and non-linear temporal kernels. The resulting analytical expressions allow us to build optimal time-series classifiers from random reservoir networks. Firstly, this allows us to optimize not only the readout vectors, but also the input projection, demonstrating a strong potential performance gain. Secondly, the analysis exposes how the second order stimulus statistics is a crucial element that interacts with the non-linearity of the dynamics and boosts performance.
Capacity of the covariance perceptron
Dec 02, 2019
The classical perceptron is a simple neural network that performs a binary classification by a linear mapping between static inputs and outputs and application of a threshold. For small inputs, neural networks in a stationary state also perform an effectively linear input-output transformation, but of an entire time series. Choosing the temporal mean of the time series as the feature for classification, the linear transformation of the network with subsequent thresholding is equivalent to the classical perceptron. Here we show that choosing covariances of time series as the feature for classification maps the neural network to what we call a 'covariance perceptron'; a bilinear mapping between covariances. By extending Gardner's theory of connections to this bilinear problem, using a replica symmetric mean-field theory, we compute the pattern and information capacities of the covariance perceptron in the infinite-size limit. Closed-form expressions reveal superior pattern capacity in the binary classification task compared to the classical perceptron in the case of a high-dimensional input and low-dimensional output. For less convergent networks, the mean perceptron classifies a larger number of stimuli. However, since covariances span a much larger input and output space than means, the amount of stored information in the covariance perceptron exceeds the classical counterpart. For strongly convergent connectivity it is superior by a factor equal to the number of input neurons. Theoretical calculations are validated numerically for finite size systems using a gradient-based optimization of a soft-margin, as well as numerical solvers for the NP hard quadratically constrained quadratic programming problem, to which training can be mapped.