# Maximum-and-Concatenation Networks

ICML 2020, 2020.

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Abstract:

While successful in many fields, deep neural networks (DNNs) still suffer from some open problems such as bad local minima and unsatisfactory generalization performance. In this work, we propose a novel architecture called Maximum-and-Concatenation Networks (MCN) to try eliminating bad local minima and improving generalization ability a...More

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Introduction

- Deep neural networks (DNNs) have been showing superior performance in various fields such as computer vision, speech recognition, natural language processing, and so on.
- Some recent theories (Zhang et al, 2017; Wei & Ma, 2019; Cao & Gu, 2019; Li & Liang, 2018; Allen-Zhu et al, 2018; Arora et al, 2019c) have revealed that, whenever the local minima produces only small training error, DNNs have probably good generalization performance at these local minima.
- While impressive, existing studies are still unsatisfactory in some aspects:

Highlights

- Deep neural networks (DNNs) have been showing superior performance in various fields such as computer vision, speech recognition, natural language processing, and so on
- Unlike the previous analyses in (Liang et al, 2018a;b; Kawaguchi & Kaelbling, 2019), which focus on the elimination of local minima but ignore the generalization performance, we provide rigorous analysis to guarantee the generalization ability of Maximum-andConcatenation Networks under certain conditions (Theorem 3 and Corollary 3.1)
- We propose a novel multi-layer Deep neural networks structure termed Maximum-andConcatenation Networks, which can approximate some class of continuous functions arbitrarily well even with highly sparse connection
- We prove that the global minima of an l-layer Maximum-andConcatenation Networks may be outperformed, at least can be attained, by increasing the network depth
- Maximum-andConcatenation Networks could be appended to any of the many existing Deep neural networks and the augmented Deep neural networks will share the same property of Maximum-andConcatenation Networks
- We analyze the generalization ability of Maximum-andConcatenation Networks and reveal that depth is more important than width for generalization; this supports the mechanism of deep learning

Methods

- The authors first construct a baseline network with 6 weighted layers, including five convolutional layers and one fully-connected layer.
- The authors add convolutional layers to make the network deeper.
- It contains five max pooling in total.
- For the MCN, the authors replace the convolutional layers after the third max pooling layer with the MCN block.
- To make a fair comparison, both networks have the same number of layers and parameters, and so for the random seed and learning rate.
- For detailed experimental settings and model configurations, please refer to the supplementary material

Results

- The authors present the main results of this paper, including a couple of theories regarding the optimality, fitting ability and generalization performance.
- All the detailed proofs of these theorems are provided in the supplementary material.
- First note that an (l + 1)-layer MCN is obtained by adding one layer into the network consisting of its first l layers, i.e., θl+1 = {θl, θ(Ll+1, Wl+1, Al+1, Al+1)}.
- Theorem 1 (Effects of Depth).
- Let the activation function γ(·) be the element-wise exp(·).
- Suppose that the loss function (·) in (3) is differentiable and convex.
- Denote by θl+1 any local minimum of an (l + 1)-layer MCN.

Conclusion

- The authors propose a novel multi-layer DNN structure termed MCN, which can approximate some class of continuous functions arbitrarily well even with highly sparse connection.
- The authors prove that the global minima of an l-layer MCN may be outperformed, at least can be attained, by increasing the network depth.
- The authors analyze the generalization ability of MCN and reveal that depth is more important than width for generalization; this supports the mechanism of deep learning.
- This study does take a step towards the ultimate goal of deep learning theory—to understand why DNNs can work well in a wide variety of applications

Summary

## Introduction:

Deep neural networks (DNNs) have been showing superior performance in various fields such as computer vision, speech recognition, natural language processing, and so on.- Some recent theories (Zhang et al, 2017; Wei & Ma, 2019; Cao & Gu, 2019; Li & Liang, 2018; Allen-Zhu et al, 2018; Arora et al, 2019c) have revealed that, whenever the local minima produces only small training error, DNNs have probably good generalization performance at these local minima.
- While impressive, existing studies are still unsatisfactory in some aspects:
## Methods:

The authors first construct a baseline network with 6 weighted layers, including five convolutional layers and one fully-connected layer.- The authors add convolutional layers to make the network deeper.
- It contains five max pooling in total.
- For the MCN, the authors replace the convolutional layers after the third max pooling layer with the MCN block.
- To make a fair comparison, both networks have the same number of layers and parameters, and so for the random seed and learning rate.
- For detailed experimental settings and model configurations, please refer to the supplementary material
## Results:

The authors present the main results of this paper, including a couple of theories regarding the optimality, fitting ability and generalization performance.- All the detailed proofs of these theorems are provided in the supplementary material.
- First note that an (l + 1)-layer MCN is obtained by adding one layer into the network consisting of its first l layers, i.e., θl+1 = {θl, θ(Ll+1, Wl+1, Al+1, Al+1)}.
- Theorem 1 (Effects of Depth).
- Let the activation function γ(·) be the element-wise exp(·).
- Suppose that the loss function (·) in (3) is differentiable and convex.
- Denote by θl+1 any local minimum of an (l + 1)-layer MCN.
## Conclusion:

The authors propose a novel multi-layer DNN structure termed MCN, which can approximate some class of continuous functions arbitrarily well even with highly sparse connection.- The authors prove that the global minima of an l-layer MCN may be outperformed, at least can be attained, by increasing the network depth.
- The authors analyze the generalization ability of MCN and reveal that depth is more important than width for generalization; this supports the mechanism of deep learning.
- This study does take a step towards the ultimate goal of deep learning theory—to understand why DNNs can work well in a wide variety of applications

- Table1: The training error (Err.) and testing accuracy (Acc.) of different models on the CIFAR-10 dataset. We denote by C the added two convolutional layers and M the appended MCN blocks
- Table2: The training error (Err.) and testing accuracy (Acc.) of different models on the CIFAR-100 dataset. We denote by C the added two convolutional layers and M the appended MCN blocks

Funding

- This work is supported in part by New Generation AI Major Project of Ministry of Science and Technology of China (grant no 2018AAA0102501), in part by NSF China (grant no.s 61625301 and 61731018), in part by Major Scientific Research Project of Zhejiang Lab (grant no.s 2019KB0AC01 and 2019KB0AB02), in part by Fundamental Research Funds of Shandong University, in part by Beijing Academy of Artificial Intelligence, in part by Qualcomm, and in part by SenseTime Research Fund

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