arXiv subjects

Jacob A. Zavatone-Veth

Publications and source records attributed to Jacob A. Zavatone-Veth.

24 records · Page 2Linked to original sources

arXiv2022

Asymptotics of representation learning in finite Bayesian neural networks

Recent works have suggested that finite Bayesian neural networks may sometimes outperform their infinite cousins because finite networks can flexibly adapt their internal representations. However, our theoretical understanding of how the learned hidden layer representations of finite networks differ from the fixed representations of infinite networks remains incomplete. Perturbative finite-width corrections to the network prior and posterior have been studied, but the asymptotics of learned features have not been fully characterized. Here, we argue that the leading finite-width corrections to the average feature kernels for any Bayesian network with linear readout and Gaussian likelihood have a largely universal form. We illustrate this explicitly for three tractable network architectures: deep linear fully-connected and convolutional networks, and networks with a single nonlinear hidden layer. Our results begin to elucidate how task-relevant learning signals shape the hidden layer representations of wide Bayesian neural networks.

cs.LG↗

arXiv2022

On neural network kernels and the storage capacity problem

In this short note, we reify the connection between work on the storage capacity problem in wide two-layer treelike neural networks and the rapidly-growing body of literature on kernel limits of wide neural networks. Concretely, we observe that the "effective order parameter" studied in the statistical mechanics literature is exactly equivalent to the infinite-width Neural Network Gaussian Process Kernel. This correspondence connects the expressivity and trainability of wide two-layer neural networks.

cond-mat.dis-nn↗

arXiv2021

Parallel locomotor control strategies in mice and flies

Our understanding of the neural basis of locomotor behavior can be informed by careful quantification of animal movement. Classical descriptions of legged locomotion have defined discrete locomotor gaits, characterized by distinct patterns of limb movement. Recent technical advances have enabled increasingly detailed characterization of limb kinematics across many species, imposing tighter constraints on neural control. Here, we highlight striking similarities between coordination patterns observed in two genetic model organisms: the laboratory mouse and Drosophila. Both species exhibit continuously-variable coordination patterns with similar low-dimensional structure, suggesting shared principles for limb coordination and descending neural control.

q-bio.NC↗

arXiv2021

Depth induces scale-averaging in overparameterized linear Bayesian neural networks

Inference in deep Bayesian neural networks is only fully understood in the infinite-width limit, where the posterior flexibility afforded by increased depth washes out and the posterior predictive collapses to a shallow Gaussian process. Here, we interpret finite deep linear Bayesian neural networks as data-dependent scale mixtures of Gaussian process predictors across output channels. We leverage this observation to study representation learning in these networks, allowing us to connect limiting results obtained in previous studies within a unified framework. In total, these results advance our analytical understanding of how depth affects inference in a simple class of Bayesian neural networks.

cs.LG↗

arXiv2021

Exact marginal prior distributions of finite Bayesian neural networks

Bayesian neural networks are theoretically well-understood only in the infinite-width limit, where Gaussian priors over network weights yield Gaussian priors over network outputs. Recent work has suggested that finite Bayesian networks may outperform their infinite counterparts, but their non-Gaussian function space priors have been characterized only though perturbative approaches. Here, we derive exact solutions for the function space priors for individual input examples of a class of finite fully-connected feedforward Bayesian neural networks. For deep linear networks, the prior has a simple expression in terms of the Meijer $G$-function. The prior of a finite ReLU network is a mixture of the priors of linear networks of smaller widths, corresponding to different numbers of active units in each layer. Our results unify previous descriptions of finite network priors in terms of their tail decay and large-width behavior.

cs.LG↗

arXiv2021

Activation function dependence of the storage capacity of treelike neural networks

The expressive power of artificial neural networks crucially depends on the nonlinearity of their activation functions. Though a wide variety of nonlinear activation functions have been proposed for use in artificial neural networks, a detailed understanding of their role in determining the expressive power of a network has not emerged. Here, we study how activation functions affect the storage capacity of treelike two-layer networks. We relate the boundedness or divergence of the capacity in the infinite-width limit to the smoothness of the activation function, elucidating the relationship between previously studied special cases. Our results show that nonlinearity can both increase capacity and decrease the robustness of classification, and provide simple estimates for the capacity of networks with several commonly used activation functions. Furthermore, they generate a hypothesis for the functional benefit of dendritic spikes in branched neurons.

cond-mat.dis-nn↗