An Analytic Solution to Covariance Propagation in Neural Networks

• URL: http://arxiv.org/abs/2403.16163v1
• Date: Sun, 24 Mar 2024 14:08:24 GMT
• Title: An Analytic Solution to Covariance Propagation in Neural Networks
• Authors: Oren Wright, Yorie Nakahira, José M. F. Moura,
• Abstract summary: This paper presents a sample-free moment propagation technique to accurately characterize the input-output distributions of neural networks. A key enabler of our technique is an analytic solution for the covariance of random variables passed through nonlinear activation functions. The wide applicability and merits of the proposed technique are shown in experiments analyzing the input-output distributions of trained neural networks and training Bayesian neural networks.
• Score: 10.013553984400488
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Uncertainty quantification of neural networks is critical to measuring the reliability and robustness of deep learning systems. However, this often involves costly or inaccurate sampling methods and approximations. This paper presents a sample-free moment propagation technique that propagates mean vectors and covariance matrices across a network to accurately characterize the input-output distributions of neural networks. A key enabler of our technique is an analytic solution for the covariance of random variables passed through nonlinear activation functions, such as Heaviside, ReLU, and GELU. The wide applicability and merits of the proposed technique are shown in experiments analyzing the input-output distributions of trained neural networks and training Bayesian neural networks.

Related papers

• Graph Neural Networks for Learning Equivariant Representations of Neural Networks [55.04145324152541]
  We propose to represent neural networks as computational graphs of parameters. Our approach enables a single model to encode neural computational graphs with diverse architectures. We showcase the effectiveness of our method on a wide range of tasks, including classification and editing of implicit neural representations.
  arXiv  Detail & Related papers  (2024-03-18T18:01:01Z)
• On the Convergence of Locally Adaptive and Scalable Diffusion-Based Sampling Methods for Deep Bayesian Neural Network Posteriors [2.3265565167163906]
  Bayesian neural networks are a promising approach for modeling uncertainties in deep neural networks. generating samples from the posterior distribution of neural networks is a major challenge. One advance in that direction would be the incorporation of adaptive step sizes into Monte Carlo Markov chain sampling algorithms. In this paper, we demonstrate that these methods can have a substantial bias in the distribution they sample, even in the limit of vanishing step sizes and at full batch size.
  arXiv  Detail & Related papers  (2024-03-13T15:21:14Z)
• Tractable Function-Space Variational Inference in Bayesian Neural Networks [72.97620734290139]
  A popular approach for estimating the predictive uncertainty of neural networks is to define a prior distribution over the network parameters. We propose a scalable function-space variational inference method that allows incorporating prior information. We show that the proposed method leads to state-of-the-art uncertainty estimation and predictive performance on a range of prediction tasks.
  arXiv  Detail & Related papers  (2023-12-28T18:33:26Z)
• Learning Theory of Distribution Regression with Neural Networks [6.961253535504979]
  We establish an approximation theory and a learning theory of distribution regression via a fully connected neural network (FNN) In contrast to the classical regression methods, the input variables of distribution regression are probability measures.
  arXiv  Detail & Related papers  (2023-07-07T09:49:11Z)
• Variational Neural Networks [88.24021148516319]
  We propose a method for uncertainty estimation in neural networks called Variational Neural Network (VNN) VNN generates parameters for the output distribution of a layer by transforming its inputs with learnable sub-layers. In uncertainty quality estimation experiments, we show that VNNs achieve better uncertainty quality than Monte Carlo Dropout or Bayes By Backpropagation methods.
  arXiv  Detail & Related papers  (2022-07-04T15:41:02Z)
• Data-driven emergence of convolutional structure in neural networks [83.4920717252233]
  We show how fully-connected neural networks solving a discrimination task can learn a convolutional structure directly from their inputs. By carefully designing data models, we show that the emergence of this pattern is triggered by the non-Gaussian, higher-order local structure of the inputs.
  arXiv  Detail & Related papers  (2022-02-01T17:11:13Z)
• An Overview of Uncertainty Quantification Methods for Infinite Neural Networks [0.0]
  We review methods for quantifying uncertainty in infinite-width neural networks. We make use of several equivalence results along the way to obtain exact closed-form solutions for predictive uncertainty.
  arXiv  Detail & Related papers  (2022-01-13T00:03:22Z)
• Multivariate Deep Evidential Regression [77.34726150561087]
  A new approach with uncertainty-aware neural networks shows promise over traditional deterministic methods. We discuss three issues with a proposed solution to extract aleatoric and epistemic uncertainties from regression-based neural networks.
  arXiv  Detail & Related papers  (2021-04-13T12:20:18Z)
• Sampling-free Variational Inference for Neural Networks with Multiplicative Activation Noise [51.080620762639434]
  We propose a more efficient parameterization of the posterior approximation for sampling-free variational inference. Our approach yields competitive results for standard regression problems and scales well to large-scale image classification tasks.
  arXiv  Detail & Related papers  (2021-03-15T16:16:18Z)
• Neural Network Solutions to Differential Equations in Non-Convex Domains: Solving the Electric Field in the Slit-Well Microfluidic Device [1.7188280334580193]
  The neural network method is used to approximate the electric potential and corresponding electric field in a slit-well microfluidic device. metrics, deep neural networks significantly outperform shallow neural networks.
  arXiv  Detail & Related papers  (2020-04-25T21:20:03Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.