On Approximation Capabilities of ReLU Activation and Softmax Output Layer in Neural Networks

• URL: http://arxiv.org/abs/2002.04060v1
• Date: Mon, 10 Feb 2020 19:48:47 GMT
• Title: On Approximation Capabilities of ReLU Activation and Softmax Output Layer in Neural Networks
• Authors: Behnam Asadi, Hui Jiang
• Abstract summary: We prove that a sufficiently large neural network using the ReLU activation function can approximate any function in $L1$ up to any arbitrary precision. We also show that a large enough neural network using a nonlinear softmax output layer can also approximate any indicator function in $L1$.
• Score: 6.852561400929072
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we have extended the well-established universal approximator theory to neural networks that use the unbounded ReLU activation function and a nonlinear softmax output layer. We have proved that a sufficiently large neural network using the ReLU activation function can approximate any function in $L^1$ up to any arbitrary precision. Moreover, our theoretical results have shown that a large enough neural network using a nonlinear softmax output layer can also approximate any indicator function in $L^1$, which is equivalent to mutually-exclusive class labels in any realistic multiple-class pattern classification problems. To the best of our knowledge, this work is the first theoretical justification for using the softmax output layers in neural networks for pattern classification.

Related papers

• Universal Consistency of Wide and Deep ReLU Neural Networks and Minimax Optimal Convergence Rates for Kolmogorov-Donoho Optimal Function Classes [7.433327915285969]
  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.
  arXiv  Detail & Related papers  (2024-01-08T23:54:46Z)
• 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)
• Exploring the Approximation Capabilities of Multiplicative Neural Networks for Smooth Functions [9.936974568429173]
  We consider two classes of target functions: generalized bandlimited functions and Sobolev-Type balls. Our results demonstrate that multiplicative neural networks can approximate these functions with significantly fewer layers and neurons. These findings suggest that multiplicative gates can outperform standard feed-forward layers and have potential for improving neural network design.
  arXiv  Detail & Related papers  (2023-01-11T17:57:33Z)
• Robust Training and Verification of Implicit Neural Networks: A Non-Euclidean Contractive Approach [64.23331120621118]
  This paper proposes a theoretical and computational framework for training and robustness verification of implicit neural networks. We introduce a related embedded network and show that the embedded network can be used to provide an $ell_infty$-norm box over-approximation of the reachable sets of the original network. We apply our algorithms to train implicit neural networks on the MNIST dataset and compare the robustness of our models with the models trained via existing approaches in the literature.
  arXiv  Detail & Related papers  (2022-08-08T03:13:24Z)
• A Local Geometric Interpretation of Feature Extraction in Deep Feedforward Neural Networks [13.159994710917022]
  In this paper, we present a local geometric analysis to interpret how deep feedforward neural networks extract low-dimensional features from high-dimensional data. Our study shows that, in a local geometric region, the optimal weight in one layer of the neural network and the optimal feature generated by the previous layer comprise a low-rank approximation of a matrix that is determined by the Bayes action of this layer.
  arXiv  Detail & Related papers  (2022-02-09T18:50:00Z)
• Neural networks with linear threshold activations: structure and algorithms [1.795561427808824]
  We show that 2 hidden layers are necessary and sufficient to represent any function representable in the class. We also give precise bounds on the sizes of the neural networks required to represent any function in the class. We propose a new class of neural networks that we call shortcut linear threshold networks.
  arXiv  Detail & Related papers  (2021-11-15T22:33:52Z)
• Why Lottery Ticket Wins? A Theoretical Perspective of Sample Complexity on Pruned Neural Networks [79.74580058178594]
  We analyze the performance of training a pruned neural network by analyzing the geometric structure of the objective function. We show that the convex region near a desirable model with guaranteed generalization enlarges as the neural network model is pruned.
  arXiv  Detail & Related papers  (2021-10-12T01:11:07Z)
• Going Beyond Linear RL: Sample Efficient Neural Function Approximation [76.57464214864756]
  We study function approximation with two-layer neural networks. Our results significantly improve upon what can be attained with linear (or eluder dimension) methods.
  arXiv  Detail & Related papers  (2021-07-14T03:03:56Z)
• 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)
• Provable Model-based Nonlinear Bandit and Reinforcement Learning: Shelve Optimism, Embrace Virtual Curvature [61.22680308681648]
  We show that global convergence is statistically intractable even for one-layer neural net bandit with a deterministic reward. For both nonlinear bandit and RL, the paper presents a model-based algorithm, Virtual Ascent with Online Model Learner (ViOL)
  arXiv  Detail & Related papers  (2021-02-08T12:41:56Z)

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.