Tutorials
The role of the tutorials is to provide a platform for a more intensive scientific exchange amongst researchers interested in a particular topic and as a meeting point for the community. Tutorials complement the depthoriented technical sessions by providing participants with broad overviews of emerging fields. A tutorial can be scheduled for 1.5 or 3 hours.
Tutorial proposals are accepted until:
August 6, 2019
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A Functional Theory Framework for Neural Networks that Includes Training Data
Lecture

Umberto Michelucci
TOELT LLC; University of Applied Science of Zürich
Switzerland


Brief Bio
Umberto studied physics and mathematics. He is an expert in numerical simulation, statistics, data science and machine learning. In addition to several years of research experience at the George Washington University (USA) and the University of Augsburg (DE), he has 15 years of practical experience in the fields of data warehouse, data science and machine learning. His last book “Applied Deep Learning – A CaseBased Approach to Understanding Deep Neural Networks” was published by Springer in 2018. He is currently working on his new book on “Convolutional and Recurrent Neural Networks Theory and Applications”. He is very active in research in the field of artificial intelligence. He publishes his research results regularly in leading journals and gives regular talks at international conferences. He teaches as lecturer at the ZHAW University of Applied Sciences for Deep Learning and Neural Networks Theory and Applications and at the HWZ University of economic science for Big Data Analysis and statistics. He is also responsible at Helsana Vesicherung AG for research and collaborations with universities in the area of AI. He just recently founded TOELT GmbH, a company aiming to develop new and modern teaching, coaching and research methods for AI, to make AI technologies and research accessible to every company and everyone.

Abstract
In this tutorial I would like to present a complete review of the functional theory framework that helps us in understanding the problem if neural networks are universal approximators and why we can use as we do. Starting from Kolmogorov's Theorem and going thruogh results from Cybenko and Hanin (the last results are from 2017) I will explain the functional theory (and in particular in terms of metric spaces) of neural networks. In particular I will highlight the differences between the theory and how in practice are neural networks trained. In particular we will look at the role of the cost function (that plays the role of the metric) and at training methods. We will also show what metrics can be used for regression and a generalised metric for classification that reduces to the classical cross entropy when used for binary classification. We will also look at what part of this framework is influenced by the training dataset and why choosing different data will give different results (apart from the obvious reasons), since changing the training data will change the metric space in which we act.
Keywords
Neural Networks; functional theory; metric spaces; universal approximation theorem; deep learning; training
Aims and Learning Objectives
The aim is to learn the fundamental theorems about neural networks that allows practitioners to use them as they do. We will learn why we can use neural networks for regression (or classification) and what hypothesis we implicitly do without mentioning them when we write programs that train neural networks. Additionally I will show a functional theory framework that will include neural network architecture and training dataset.
Target Audience
Practitioners interested in the mathematical theory of neural networks. Mathematicians that study the problem of learning and computer scientist that wants to understand the fundamentals properties of neural networks.
Prerequisite Knowledge of Audience
Basic understanding of Analysis and mathematics at the undergraduate level. I will cover the basics of functional theory at the beginning. Basic understanding of how neural networks are working is an advantage. But I will also briefly cover that in the beginning.
Detailed Outline
 Neural network introduction (a more mathematical view)
 Neural network introduction (a programming view with code snippets in Tensorflow and Python)
 Functional theory introduction
 Review of the theory from the 80s to today of the theory that describe the problem of under which conditions are network universal approximators.
 Description of the framework for neural networks and some new metric that allows to discuss about regression and classification in terms of metric spaces