Learning Theory of Distribution Regression with Neural Networks

• URL: http://arxiv.org/abs/2307.03487v1
• Date: Fri, 7 Jul 2023 09:49:11 GMT
• Title: Learning Theory of Distribution Regression with Neural Networks
• Authors: Zhongjie Shi, Zhan Yu, Ding-Xuan Zhou
• Abstract summary: 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.
• Score: 6.961253535504979
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we aim at establishing 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. Then we often need to perform a second-stage sampling process to approximate the actual information of the distribution. On the other hand, the classical neural network structure requires the input variable to be a vector. When the input samples are probability distributions, the traditional deep neural network method cannot be directly used and the difficulty arises for distribution regression. A well-defined neural network structure for distribution inputs is intensively desirable. There is no mathematical model and theoretical analysis on neural network realization of distribution regression. To overcome technical difficulties and address this issue, we establish a novel fully connected neural network framework to realize an approximation theory of functionals defined on the space of Borel probability measures. Furthermore, based on the established functional approximation results, in the hypothesis space induced by the novel FNN structure with distribution inputs, almost optimal learning rates for the proposed distribution regression model up to logarithmic terms are derived via a novel two-stage error decomposition technique.

Related papers

• Novel Kernel Models and Exact Representor Theory for Neural Networks Beyond the Over-Parameterized Regime [52.00917519626559]
  This paper presents two models of neural-networks and their training applicable to neural networks of arbitrary width, depth and topology. We also present an exact novel representor theory for layer-wise neural network training with unregularized gradient descent in terms of a local-extrinsic neural kernel (LeNK) This representor theory gives insight into the role of higher-order statistics in neural network training and the effect of kernel evolution in neural-network kernel models.
  arXiv  Detail & Related papers  (2024-05-24T06:30:36Z)
• An Analytic Solution to Covariance Propagation in Neural Networks [10.013553984400488]
  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.
  arXiv  Detail & Related papers  (2024-03-24T14:08:24Z)
• Deep Neural Networks Tend To Extrapolate Predictably [51.303814412294514]
  neural network predictions tend to be unpredictable and overconfident when faced with out-of-distribution (OOD) inputs. We observe that neural network predictions often tend towards a constant value as input data becomes increasingly OOD. We show how one can leverage our insights in practice to enable risk-sensitive decision-making in the presence of OOD inputs.
  arXiv  Detail & Related papers  (2023-10-02T03:25:32Z)
• Semantic Strengthening of Neuro-Symbolic Learning [85.6195120593625]
  Neuro-symbolic approaches typically resort to fuzzy approximations of a probabilistic objective. We show how to compute this efficiently for tractable circuits. We test our approach on three tasks: predicting a minimum-cost path in Warcraft, predicting a minimum-cost perfect matching, and solving Sudoku puzzles.
  arXiv  Detail & Related papers  (2023-02-28T00:04:22Z)
• Mean-Field Analysis of Two-Layer Neural Networks: Global Optimality with Linear Convergence Rates [7.094295642076582]
  Mean-field regime is a theoretically attractive alternative to the NTK (lazy training) regime. We establish a new linear convergence result for two-layer neural networks trained by continuous-time noisy descent in the mean-field regime.
  arXiv  Detail & Related papers  (2022-05-19T21:05:40Z)
• Approximation bounds for norm constrained neural networks with applications to regression and GANs [9.645327615996914]
  We prove upper and lower bounds on the approximation error of ReLU neural networks with norm constraint on the weights. We apply these approximation bounds to analyze the convergences of regression using norm constrained neural networks and distribution estimation by GANs.
  arXiv  Detail & Related papers  (2022-01-24T02:19:05Z)
• PAC-Bayesian Learning of Aggregated Binary Activated Neural Networks with Probabilities over Representations [2.047424180164312]
  We study the expectation of a probabilistic neural network as a predictor by itself, focusing on the aggregation of binary activated neural networks with normal distributions over real-valued weights. We show that the exact computation remains tractable for deep but narrow neural networks, thanks to a dynamic programming approach.
  arXiv  Detail & Related papers  (2021-10-28T14:11:07Z)
• FF-NSL: Feed-Forward Neural-Symbolic Learner [70.978007919101]
  This paper introduces a neural-symbolic learning framework, called Feed-Forward Neural-Symbolic Learner (FF-NSL) FF-NSL integrates state-of-the-art ILP systems based on the Answer Set semantics, with neural networks, in order to learn interpretable hypotheses from labelled unstructured data.
  arXiv  Detail & Related papers  (2021-06-24T15:38:34Z)
• Improving Uncertainty Calibration via Prior Augmented Data [56.88185136509654]
  Neural networks have proven successful at learning from complex data distributions by acting as universal function approximators. They are often overconfident in their predictions, which leads to inaccurate and miscalibrated probabilistic predictions. We propose a solution by seeking out regions of feature space where the model is unjustifiably overconfident, and conditionally raising the entropy of those predictions towards that of the prior distribution of the labels.
  arXiv  Detail & Related papers  (2021-02-22T07:02:37Z)
• Measurement error models: from nonparametric methods to deep neural networks [3.1798318618973362]
  We propose an efficient neural network design for estimating measurement error models. We use a fully connected feed-forward neural network to approximate the regression function $f(x)$. We conduct an extensive numerical study to compare the neural network approach with classical nonparametric methods.
  arXiv  Detail & Related papers  (2020-07-15T06:05:37Z)
• Network Diffusions via Neural Mean-Field Dynamics [52.091487866968286]
  We propose a novel learning framework for inference and estimation problems of diffusion on networks. Our framework is derived from the Mori-Zwanzig formalism to obtain an exact evolution of the node infection probabilities. Our approach is versatile and robust to variations of the underlying diffusion network models.
  arXiv  Detail & Related papers  (2020-06-16T18:45:20Z)

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.