Universal Consistency of Wide and Deep ReLU Neural Networks and Minimax Optimal Convergence Rates for Kolmogorov-Donoho Optimal Function Classes

• URL: http://arxiv.org/abs/2401.04286v2
• Date: Tue, 30 Jan 2024 23:12:06 GMT
• Title: Universal Consistency of Wide and Deep ReLU Neural Networks and Minimax Optimal Convergence Rates for Kolmogorov-Donoho Optimal Function Classes
• Authors: Hyunouk Ko and Xiaoming Huo
• Abstract summary: We prove the universal consistency of wide and deep ReLU neural network classifiers trained on the logistic loss. We also give sufficient conditions for a class of probability measures for which classifiers based on neural networks achieve minimax optimal rates of convergence.
• Score: 7.433327915285969
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we prove the universal consistency of wide and deep ReLU neural network classifiers trained on the logistic loss. We also give sufficient conditions for a class of probability measures for which classifiers based on neural networks achieve minimax optimal rates of convergence. The result applies to a wide range of known function classes. In particular, while most previous works impose explicit smoothness assumptions on the regression function, our framework encompasses more general settings. The proposed neural networks are either the minimizers of the logistic loss or the $0$-$1$ loss. In the former case, they are interpolating classifiers that exhibit a benign overfitting behavior.

Related papers

• On Excess Risk Convergence Rates of Neural Network Classifiers [8.329456268842227]
  We study the performance of plug-in classifiers based on neural networks in a binary classification setting as measured by their excess risks. We analyze the estimation and approximation properties of neural networks to obtain a dimension-free, uniform rate of convergence.
  arXiv  Detail & Related papers  (2023-09-26T17:14:10Z)
• Does a sparse ReLU network training problem always admit an optimum? [0.0]
  We show that the existence of an optimal solution is not always guaranteed, especially in the context of sparse ReLU neural networks. In particular, we first show that optimization problems involving deep networks with certain sparsity patterns do not always have optimal parameters.
  arXiv  Detail & Related papers  (2023-06-05T08:01:50Z)
• Benign Overfitting in Deep Neural Networks under Lazy Training [72.28294823115502]
  We show that when the data distribution is well-separated, DNNs can achieve Bayes-optimal test error for classification. Our results indicate that interpolating with smoother functions leads to better generalization.
  arXiv  Detail & Related papers  (2023-05-30T19:37:44Z)
• Globally Optimal Training of Neural Networks with Threshold Activation Functions [63.03759813952481]
  We study weight decay regularized training problems of deep neural networks with threshold activations. We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
  arXiv  Detail & Related papers  (2023-03-06T18:59:13Z)
• Optimal Learning Rates of Deep Convolutional Neural Networks: Additive Ridge Functions [19.762318115851617]
  We consider the mean squared error analysis for deep convolutional neural networks. We show that, for additive ridge functions, convolutional neural networks followed by one fully connected layer with ReLU activation functions can reach optimal mini-max rates.
  arXiv  Detail & Related papers  (2022-02-24T14:22:32Z)
• Critical Initialization of Wide and Deep Neural Networks through Partial Jacobians: General Theory and Applications [6.579523168465526]
  We introduce emphpartial Jacobians of a network, defined as derivatives of preactivations in layer $l$ with respect to preactivations in layer $l_0leq l$. We derive recurrence relations for the norms of partial Jacobians and utilize these relations to analyze criticality of deep fully connected neural networks with LayerNorm and/or residual connections.
  arXiv  Detail & Related papers  (2021-11-23T20:31:42Z)
• Non-Gradient Manifold Neural Network [79.44066256794187]
  Deep neural network (DNN) generally takes thousands of iterations to optimize via gradient descent. We propose a novel manifold neural network based on non-gradient optimization.
  arXiv  Detail & Related papers  (2021-06-15T06:39:13Z)
• Towards an Understanding of Benign Overfitting in Neural Networks [104.2956323934544]
  Modern machine learning models often employ a huge number of parameters and are typically optimized to have zero training loss. We examine how these benign overfitting phenomena occur in a two-layer neural network setting. We show that it is possible for the two-layer ReLU network interpolator to achieve a near minimax-optimal learning rate.
  arXiv  Detail & Related papers  (2021-06-06T19:08:53Z)
• A Geometric Analysis of Neural Collapse with Unconstrained Features [40.66585948844492]
  We provide the first global optimization landscape analysis of $Neural;Collapse$. This phenomenon arises in the last-layer classifiers and features of neural networks during the terminal phase of training.
  arXiv  Detail & Related papers  (2021-05-06T00:00:50Z)
• Topological obstructions in neural networks learning [67.8848058842671]
  We study global properties of the loss gradient function flow. We use topological data analysis of the loss function and its Morse complex to relate local behavior along gradient trajectories with global properties of the loss surface.
  arXiv  Detail & Related papers  (2020-12-31T18:53:25Z)
• The Hidden Convex Optimization Landscape of Two-Layer ReLU Neural Networks: an Exact Characterization of the Optimal Solutions [51.60996023961886]
  We prove that finding all globally optimal two-layer ReLU neural networks can be performed by solving a convex optimization program with cone constraints. Our analysis is novel, characterizes all optimal solutions, and does not leverage duality-based analysis which was recently used to lift neural network training into convex spaces.
  arXiv  Detail & Related papers  (2020-06-10T15:38:30Z)

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.