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Illustrative introductions on dimension reduction

“What is your image on dimensions?”

….That might be a cheesy question to ask to reader of Data Science Blog, but most people, with no scientific background, would answer “One dimension is a line, and two dimension is a plain, and we live in three-dimensional world.” After that if you ask “How about the fourth dimension?” many people would answer “Time?”

You can find books or writings about dimensions in various field. And you can use the word “dimension” in normal conversations, in many contexts.

*In Japanese, if you say “He likes two dimension.” that means he prefers anime characters to real women, as is often the case with Japanese computer science students.

The meanings of “dimensions” depend on the context, but in data science dimension is usually the number of rows of your Excel data.

When you study data science or machine learning, usually you should start with understanding the algorithms with 2 or 3 dimensional data, and you can apply those ideas to any D dimensional data. But of course you cannot visualize D dimensional data anymore, and you always have to be careful of what happens if you expand degree of dimension.

Conversely it is also important to reduce dimension to understand abstract high dimensional stuff in 2 or 3 dimensional space, which are close to our everyday sense. That means dimension reduction is one powerful way of data visualization.

In this blog series I am going to explain meanings of dimension itself in machine learning context and algorithms for dimension reductions, such as PCA, LDA, and t-SNE, with 2 or 3 dimensional visible data. Along with that, I am going to delve into the meaning of calculations so that you can understand them in more like everyday-life sense.

This article series is going to be roughly divided into the contents below.

  1. Curse of Dimensionality
  2. PCA, LDA (to be published soon)
  3. Rethinking eigen vectors (to be published soon)
  4. KL expansion and subspace method (to be published soon)
  5. Autoencoder as dimension reduction (to be published soon)
  6. t-SNE (to be published soon)

I hope you could see that reducing dimension is one of the fundamental approaches in data science or machine learning.

Simple RNN

Understanding LSTM forward propagation in two ways

*This article is only for the sake of understanding the equations in the second page of the paper named “LSTM: A Search Space Odyssey”. If you have no trouble understanding the equations of LSTM forward propagation, I recommend you to skip this article and go the the next article.

*This article is the fourth article of “A gentle introduction to the tiresome part of understanding RNN.”

1. Preface

I  heard that in Western culture, smart people write textbooks so that other normal people can understand difficult stuff, and that is why textbooks in Western countries tend to be bulky, but also they are not so difficult as they look. On the other hand in Asian culture, smart people write puzzling texts on esoteric topics, and normal people have to struggle to understand what noble people wanted to say. Publishers also require the authors to keep the texts as short as possible, so even though the textbooks are thin, usually students have to repeat reading the textbooks several times because usually they are too abstract.

Both styles have cons and pros, and usually I prefer Japanese textbooks because they are concise, and sometimes it is annoying to read Western style long texts with concrete straightforward examples to reach one conclusion. But a problem is that when it comes to explaining LSTM, almost all the text books are like Asian style ones. Every study material seems to skip the proper steps necessary for “normal people” to understand its algorithms. But after actually making concrete slides on mathematics on LSTM, I understood why: if you write down all the equations on LSTM forward/back propagation, that is going to be massive, and actually I had to make 100-page PowerPoint animated slides to make it understandable to people like me.

I already had a feeling that “Does it help to understand only LSTM with this precision? I should do more practical codings.” For example François Chollet, the developer of Keras, in his book, said as below.

 

For me that sounds like “We have already implemented RNNs for you, so just shut up and use Tensorflow/Keras.” Indeed, I have never cared about the architecture of my Mac Book Air, but I just use it every day, so I think he is to the point. To make matters worse, for me, a promising algorithm called Transformer seems to be replacing the position of LSTM in natural language processing. But in this article series and in my PowerPoint slides, I tried to explain as much as possible, contrary to his advice.

But I think, or rather hope,  it is still meaningful to understand this 23-year-old algorithm, which is as old as me. I think LSTM did build a generation of algorithms for sequence data, and actually Sepp Hochreiter, the inventor of LSTM, has received Neural Network Pioneer Award 2021 for his work.

I hope those who study sequence data processing in the future would come to this article series, and study basics of RNN just as I also study classical machine learning algorithms.

 *In this article “Densely Connected Layers” is written as “DCL,” and “Convolutional Neural Network” as “CNN.”

2. Why LSTM?

First of all, let’s take a brief look at what I said about the structures of RNNs,  in the first and the second article. A simple RNN is basically densely connected network with a few layers. But the RNN gets an input every time step, and it gives out an output at the time step. Part of information in the middle layer are succeeded to the next time step, and in the next time step, the RNN also gets an input and gives out an output. Therefore, virtually a simple RNN behaves almost the same way as densely connected layers with many layers during forward/back propagation if you focus on its recurrent connections.

That is why simple RNNs suffer from vanishing/exploding gradient problems, where the information exponentially vanishes or explodes when its gradients are multiplied many times through many layers during back propagation. To be exact, I think you need to consider this problem precisely like you can see in this paper. But for now, please at least keep it in mind that when you calculate a gradient of an error function with respect to parameters of simple neural networks, you have to multiply parameters many times like below, and this type of calculation usually leads to vanishing/exploding gradient problem.

LSTM was invented as a way to tackle such problems as I mentioned in the last article.

3. How to display LSTM

I would like you to just go to image search on Google, Bing, or Yahoo!, and type in “LSTM.” I think you will find many figures, but basically LSTM charts are roughly classified into two types: in this article I call them “Space Odyssey type” and “electronic circuit type”, and in conclusion, I highly recommend you to understand LSTM as the “electronic circuit type.”

*I just randomly came up with the terms “Space Odyssey type” and “electronic circuit type” because the former one is used in the paper I mentioned, and the latter one looks like an electronic circuit to me. You do not have to take how I call them seriously.

However, not that all the well-made explanations on LSTM use the “electronic circuit type,” and I am sure you sometimes have to understand LSTM as the “space odyssey type.” And the paper “LSTM: A Search Space Odyssey,” which I learned a lot about LSTM from,  also adopts the “Space Odyssey type.”

LSTM architectur visualization

The main reason why I recommend the “electronic circuit type” is that its behaviors look closer to that of simple RNNs, which you would have seen if you read my former articles.

*Behaviors of both of them look different, but of course they are doing the same things.

If you have some understanding on DCL, I think it was not so hard to understand how simple RNNs work because simple RNNs  are mainly composed of linear connections of neurons and weights, whose structures are the same almost everywhere. And basically they had only straightforward linear connections as you can see below.

But from now on, I would like you to give up the ideas that LSTM is composed of connections of neurons like the head image of this article series. If you do that, I think that would be chaotic and I do not want to make a figure of it on Power Point. In short, sooner or later you have to understand equations of LSTM.

4. Forward propagation of LSTM in “electronic circuit type”

*For further understanding of mathematics of LSTM forward/back propagation, I recommend you to download my slides.

The behaviors of an LSTM block is quite similar to that of a simple RNN block: an RNN block gets an input every time step and gets information from the RNN block of the last time step, via recurrent connections. And the block succeeds information to the next block.

Let’s look at the simplified architecture of  an LSTM block. First of all, you should keep it in mind that LSTM have two streams of information: the one going through all the gates, and the one going through cell connections, the “highway” of LSTM block. For simplicity, we will see the architecture of an LSTM block without peephole connections, the lines in blue. The flow of information through cell connections is relatively uninterrupted. This helps LSTMs to retain information for a long time.

In a LSTM block, the input and the output of the former time step separately go through sections named “gates”: input gate, forget gate, output gate, and block input. The outputs of the forget gate, the input gate, and the block input join the highway of cell connections to renew the value of the cell.

*The small two dots on the cell connections are the “on-ramp” of cell conection highway.

*You would see the terms “input gate,” “forget gate,” “output gate” almost everywhere, but how to call the “block gate” depends on textbooks.

Let’s look at the structure of an LSTM block a bit more concretely. An LSTM block at the time step (t) gets \boldsymbol{y}^{(t-1)}, the output at the last time step,  and \boldsymbol{c}^{(t-1)}, the information of the cell at the time step (t-1), via recurrent connections. The block at time step (t) gets the input \boldsymbol{x}^{(t)}, and it separately goes through each gate, together with \boldsymbol{y}^{(t-1)}. After some calculations and activation, each gate gives out an output. The outputs of the forget gate, the input gate, the block input, and the output gate are respectively \boldsymbol{f}^{(t)}, \boldsymbol{i}^{(t)}, \boldsymbol{z}^{(t)}, \boldsymbol{o}^{(t)}. The outputs of the gates are mixed with \boldsymbol{c}^{(t-1)} and the LSTM block gives out an output \boldsymbol{y}^{(t)}, and gives \boldsymbol{y}^{(t)} and \boldsymbol{c}^{(t)} to the next LSTM block via recurrent connections.

You calculate \boldsymbol{f}^{(t)}, \boldsymbol{i}^{(t)}, \boldsymbol{z}^{(t)}, \boldsymbol{o}^{(t)} as below.

  • \boldsymbol{f}^{(t)}= \sigma(\boldsymbol{W}_{for} \boldsymbol{x}^{(t)} + \boldsymbol{R}_{for} \boldsymbol{y}^{(t-1)} +  \boldsymbol{b}_{for})
  • \boldsymbol{i}^{(t)}=\sigma(\boldsymbol{W}_{in} \boldsymbol{x}^{(t)} + \boldsymbol{R}_{in} \boldsymbol{y}^{(t-1)} + \boldsymbol{b}_{in})
  • \boldsymbol{z}^{(t)}=tanh(\boldsymbol{W}_z \boldsymbol{x}^{(t)} + \boldsymbol{R}_z \boldsymbol{y}^{(t-1)} + \boldsymbol{b}_z)
  • \boldsymbol{o}^{(t)}=\sigma(\boldsymbol{W}_{out} \boldsymbol{x}^{(t)} + \boldsymbol{R}_{out} \boldsymbol{y}^{(t-1)} + \boldsymbol{b}_{out})

*You have to keep it in mind that the equations above do not include peephole connections, which I am going to show with blue lines in the end.

The equations above are quite straightforward if you understand forward propagation of simple neural networks. You add linear products of \boldsymbol{y}^{(t)} and \boldsymbol{c}^{(t)} with different weights in each gate. What makes LSTMs different from simple RNNs is how to mix the outputs of the gates with the cell connections. In order to explain that, I need to introduce a mathematical operator called Hadamard product, which you denote as \odot. This is a very simple operator. This operator produces an elementwise product of two vectors or matrices with identical shape.

With this Hadamar product operator, the renewed cell and the output are calculated as below.

  • \boldsymbol{c}^{(t)} = \boldsymbol{z}^{(t)}\odot \boldsymbol{i}^{(t)} + \boldsymbol{c}^{(t-1)} \odot \boldsymbol{f}^{(t)}
  • \boldsymbol{y}^{(t)} = \boldsymbol{o}^{(t)} \odot tanh(\boldsymbol{c}^{(t)})

The values of \boldsymbol{f}^{(t)}, \boldsymbol{i}^{(t)}, \boldsymbol{z}^{(t)}, \boldsymbol{o}^{(t)} are compressed into the range of [0, 1] or [-1, 1] with activation functions. You can see that the input gate and the block input give new information to the cell. The part \boldsymbol{c}^{(t-1)} \odot \boldsymbol{f}^{(t)} means that the output of the forget gate “forgets” the cell of the last time step by multiplying the values from 0 to 1 elementwise. And the cell \boldsymbol{c}^{(t)} is activated with tanh() and the output of the output gate “suppress” the activated value of \boldsymbol{c}^{(t)}. In other words, the output gatedecides how much information to give out as an output of the LSTM block. The output of every gate depends on the input \boldsymbol{x}^{(t)}, and the recurrent connection \boldsymbol{y}^{(t-1)}. That means an LSTM block learns to forget the cell of the last time step, to renew the cell, and to suppress the output. To describe in an extreme manner, if all the outputs of every gate are always (1, 1, …1)^T, LSTMs forget nothing, retain information of inputs at every time step, and gives out everything. And  if all the outputs of every gate are always (0, 0, …0)^T, LSTMs forget everything, receive no inputs, and give out nothing.

This model has one problem: the outputs of each gate do not directly depend on the information in the cell. To solve this problem, some LSTM models introduce some flows of information from the cell to each gate, which are shown as lines in blue in the figure below.

LSTM inner architecture

LSTM models, for example the one with or without peephole connection, depend on the library you use, and the model I have showed is one of standard LSTM structure. However no matter how complicated structure of an LSTM block looks, you usually cover it with a black box as below and show its behavior in a very simplified way.

5. Space Odyssey type

I personally think there is no advantages of understanding how LSTMs work with this Space Odyssey type chart, but in several cases you would have to use this type of chart. So I will briefly explain how to look at that type of chart, based on understandings of LSTMs you have gained through this article.

In Space Odyssey type of LSTM chart, at the center is a cell. Electronic circuit type of chart, which shows the flow of information of the cell as an uninterrupted “highway” in an LSTM block. On the other hand, in a Spacey Odyssey type of chart, the information of the cell rotate at the center. And each gate gets the information of the cell through peephole connections,  \boldsymbol{x}^{(t)}, the input at the time step (t) , sand \boldsymbol{y}^{(t-1)}, the output at the last time step (t-1), which came through recurrent connections. In Space Odyssey type of chart, you can more clearly see that the information of the cell go to each gate through the peephole connections in blue. Each gate calculates its output.

Just as the charts you have seen, the dotted line denote the information from the past. First, the information of the cell at the time step (t-1) goes to the forget gate and get mixed with the output of the forget cell In this process the cell is partly “forgotten.” Next, the input gate and the block input are mixed to generate part of new value of the the cell at time step  (t). And the partly “forgotten” \boldsymbol{c}^{(t-1)} goes back to the center of the block and it is mixed with the output of the input gate and the block input. That is how \boldsymbol{c}^{(t)} is renewed. And the value of new cell flow to the top of the chart, being mixed with the output of the output gate. Or you can also say the information of new cell is “suppressed” with the output gate.

I have finished the first four articles of this article series, and finally I am gong to write about back propagation of LSTM in the next article. I have to say what I have written so far is all for the next article, and my long long Power Point slides.

 

* I make study materials on machine learning, sponsored by DATANOMIQ. I do my best to make my content as straightforward but as precise as possible. I include all of my reference sources. If you notice any mistakes in my materials, including grammatical errors, please let me know (email: yasuto.tamura@datanomiq.de). And if you have any advice for making my materials more understandable to learners, I would appreciate hearing it.

[References]

[1] Klaus Greff, Rupesh Kumar Srivastava, Jan Koutník, Bas R. Steunebrink, Jürgen Schmidhuber, “LSTM: A Search Space Odyssey,” (2017)

[2] Francois Chollet, Deep Learning with Python,(2018), Manning , pp. 202-204

[3] “Sepp Hochreiter receives IEEE CIS Neural Networks Pioneer Award 2021”, Institute of advanced research in artificial intelligence, (2020)
URL: https://www.iarai.ac.at/news/sepp-hochreiter-receives-ieee-cis-neural-networks-pioneer-award-2021/?fbclid=IwAR27cwT5MfCw4Tqzs3MX_W9eahYDcIFuoGymATDR1A-gbtVmDpb8ExfQ87A

[4] Oketani Takayuki, “Machine Learning Professional Series: Deep Learning,” (2015), pp. 120-125
岡谷貴之 著, 「機械学習プロフェッショナルシリーズ 深層学習」, (2015), pp. 120-125

[5] Harada Tatsuya, “Machine Learning Professional Series: Image Recognition,” (2017), pp. 252-257
原田達也 著, 「機械学習プロフェッショナルシリーズ 画像認識」, (2017), pp. 252-257

[6] “Understandable LSTM ~ With the Current Trends,” Qiita, (2015)
「わかるLSTM ~ 最近の動向と共に」, Qiita, (2015)
URL: https://qiita.com/t_Signull/items/21b82be280b46f467d1b

Simple RNN

A brief history of neural nets: everything you should know before learning LSTM

This series is not a college course or something on deep learning with strict deadlines for assignments, so let’s take a detour from practical stuff and take a brief look at the history of neural networks.

The history of neural networks is also a big topic, which could be so long that I had to prepare another article series. And usually I am supposed to begin such articles with something like “The term ‘AI’ was first used by John McCarthy in Dartmouth conference 1956…” but you can find many of such texts written by people with much more experiences in this field. Therefore I am going to write this article from my point of view, as an intern writing articles on RNN, as a movie buff, and as one of many Japanese men who spent a great deal of childhood with video games.

We are now in the third AI boom, and some researchers say this boom began in 2006. A professor in my university said there we are now in a kind of bubble economy in machine learning/data science industry, but people used to say “Stop daydreaming” to AI researchers. The second AI winter is partly due to vanishing/exploding gradient problem of deep learning. And LSTM was invented as one way to tackle such problems, in 1997.

1, First AI boom

In the first AI boom, I think people were literally “daydreaming.” Even though the applications of machine learning algorithms were limited to simple tasks like playing chess, checker, or searching route of 2d mazes, and sometimes this time is called GOFAI (Good Old Fashioned AI).

Source: https://www.youtube.com/watch?v=K-HfpsHPmvw&feature=youtu.be

Even today when someone use the term “AI” merely for tasks with neural networks, that amuses me because for me deep learning is just statistically and automatically training neural networks, which are capable of universal approximation, into some classifiers/regressors. Actually the algorithms behind that is quite impressive, but the structure of human brains is much more complicated. The hype of “AI” already started in this first AI boom. Let me take an example of machine translation in this video. In fact the research of machine translation already started in the early 1950s, and of  specific interest in the time was translation between English and Russian due to Cold War. In the first article of this series, I said one of the most famous applications of RNN is machine translation, such as Google Translation, DeepL. They are a type of machine translation called neural machine translation because they use neural networks, especially RNNs. Neural machine translation was an astonishing breakthrough around 2014 in machine translation field. The former major type of machine translation was statistical machine translation, based on statistical language models. And the machine translator in the first AI boom was rule base machine translators, which are more primitive than statistical ones.

Source: https://news.cornell.edu/stories/2019/09/professors-perceptron-paved-way-ai-60-years-too-soon

The most remarkable invention in this time was of course perceptron by Frank Rosenblatt. Some people say that this is the first neural network. Even though you can implement perceptron with a-few-line codes in Python, obviously they did not have Jupyter Notebook in those days. The perceptron was implemented as a huge instrument named Mark 1 Perceptron, and it was composed of randomly connected wires. I do not precisely know how it works, but it was a huge effort to implement even the most primitive type of neural networks. They needed to use a big lighting fixture to get a 20*20 pixel image using 20*20 array of cadmium sulphide photocells. The research by Rosenblatt, however, was criticized by Marvin Minsky in his book because perceptrons could only be used for linearly separable data. To make matters worse the criticism prevailed as that more general, multi-layer perceptrons were also not useful for linearly inseparable data (as I mentioned in the first article, multi-layer perceptrons, namely normal neural networks,  can be universal approximators, which have potentials to classify/regress various types of complex data). In case you do not know what “linearly separable” means, imagine that there are data plotted on a piece of paper. If an elementary school kid can draw a border line between two clusters of the data with a ruler and a pencil on the paper, the 2d data is “linearly separable”….

With big disappointments to the research on “electronic brains,” the budget of AI research was reduced and AI research entered its first winter.

Source: https://www.nzz.ch/digital/ehre-fuer-die-deep-learning-mafia-ld.1472761?reduced=true and https://anatomiesofintelligence.github.io/posts/2019-06-21-organization-mark-i-perceptron

I think  the frame problem (1969),  by John McCarthy and Patrick J. Hayes, is also an iconic theory in the end of the first AI boom. This theory is known as a story of creating a robot trying to pull out its battery on a wheeled wagon in a room. But there is also a time bomb on the wagon. The first prototype of the robot, named R1, naively tried to pull out the wagon form the room, and the bomb exploded. The problems was obvious: R1 was not programmed to consider the risks by taking each action, so the researchers made the next prototype named R1D1, which was programmed to consider the potential risks of taking each action. When R1D1 tried to pull out the wagon, it realized the risk of pulling the bomb together with the battery. But soon it started considering all the potential risks, such as the risk of the ceiling falling down, the distance between the wagon and all the walls, and so on, when the bomb exploded. The next problem was also obvious: R1D1 was not programmed to distinguish if the factors are relevant of irrelevant to the main purpose, and the next prototype R2D1 was programmed to do distinguish them. This time, R2D1 started thinking about “whether the factor is  irrelevant to the main purpose,” on every factor measured, and again the bomb exploded. How can we get a perfect AI, R2D2?

The situation of mentioned above is a bit extreme, but it is said AI could also get stuck when it try to take some super simple actions like finding a number in a phone book and make a phone call. It is difficult for an artificial intelligence to decide what is relevant and what is irrelevant, but humans will not get stuck with such simple stuff, and sometimes the frame problem is counted as the most difficult and essential problem of developing AI. But personally I think the original frame problem was unreasonable in that McCarthy, in his attempts to model the real world, was inflexible in his handling of the various equations involved, treating them all with equal weight regardless of the particular circumstances of a situation. Some people say that McCarthy, who was an advocate for AI, also wanted to see the field come to an end, due to its failure to meet the high expectations it once aroused.

Not only the frame problem, but also many other AI-related technological/philosophical problems have been proposed, such as Chinese room (1980), the symbol grounding problem (1990), and they are thought to be as hardships in inventing artificial intelligence, but I omit those topics in this article.

*The name R2D2 did not come from the famous story of frame problem. The story was Daniel Dennett first proposed the story of R2D2 in his paper published in 1984. Star Wars was first released in 1977. It is said that the name R2D2 came from “Reel 2, Dialogue 2,” which George Lucas said while film shooting. And the design of C3PO came from Maria in Metropolis(1927). It is said that the most famous AI duo in movie history was inspired by Tahei and Matashichi in The Hidden Fortress (1958), directed by Kurosawa Akira.

Source: https://criterioncollection.tumblr.com/post/135392444906/the-original-r2-d2-and-c-3po-the-hidden-fortress

Interestingly, in the end of the first AI boom, 2001: A Space Odyssey, directed by Stanley Kubrick, was released in 1968. Unlike conventional fantasylike AI characters, for example Maria in Metropolis (1927), HAL 9000 was portrayed as a very realistic AI, and the movie already pointed out the risk of AI being insane when it gets some commands from several users. HAL 9000 still has been a very iconic character in AI field. For example when you say some quotes from 2001: A Space Odyssey to Siri you get some parody responses. I also thin you should keep it in mind that in order to make an AI like HAL 9000 come true, for now RNNs would be indispensable in many ways: you would need RNNs for better voice recognition, better conversational system, and for reading lips.

Source: https://imgflip.com/memetemplate/34339860/Open-the-pod-bay-doors-Hal

*Just as you cannot understand Monty Python references in Python official tutorials without watching Monty Python and the Holy Grail, you cannot understand many parodies in AI contexts without watching 2001: A Space Odyssey. Even though the movie had some interview videos with some researchers and some narrations, Stanley Kubrick cut off all the footage and made the movie very difficult to understand. Most people did not or do not understand that it is a movie about aliens who gave homework of coming to Jupiter to human beings.

2, Second AI boom/winter

Source: Fukushima Kunihiko, “Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position,” (1980)

I am not going to write about the second AI boom in detail, but at least you should keep it in mind that convolutional neural network (CNN) is a keyword in this time. Neocognitron, an artificial model of how sight nerves perceive thing, was invented by Kunihiko Fukushima in 1980, and the model is said to be the origin on CNN. And Neocognitron got inspired by the Hubel and Wiesel’s research on sight nerves. In 1989, a group in AT & T Bell Laboratory led by Yann LeCun invented the first practical CNN to read handwritten digit.

Y. LeCun, “Backpropagation Applied to Handwritten Zip Code Recognition,” (1989)

Another turning point in this second AI boom was that back propagation algorithm was discovered, and the CNN by LeCun was also trained with back propagation. LeCun made a deep neural networks with some layers in 1998 for more practical uses.

But his research did not gain so much attention like today, because AI research entered its second winter at the beginning of the 1990s, and that was partly due to vanishing/exploding gradient problem of deep learning. People knew that neural networks had potentials of universal approximation, but when they tried to train naively stacked neural nets, the gradients, which you need for training neural networks, exponentially increased/decreased. Even though the CNN made by LeCun was the first successful case of “deep” neural nets which did not suffer from the vanishing/exploding gradient problem so much, deep learning research also stagnated in this time.

The ultimate goal of this article series is to understand LSTM at a more abstract/mathematical level because it is one of the practical RNNs, but the idea of LSTM (Long Short Term Memory) itself was already proposed in 1997 as an RNN algorithm to tackle vanishing gradient problem. (Exploding gradient problem is solved with a technique named gradient clipping, and this is easier than techniques for preventing vanishing gradient problems. I am also going to explain it in the next article.) After that some other techniques like introducing forget gate, peephole connections, were discovered, but basically it took some 20 years till LSTM got attentions like today. The reasons for that is lack of hardware and data sets, and that was also major reasons for the second AI winter.

Source: Sepp HochreiterJürgen, Schmidhuber, “Long Short-term Memory,” (1997)

In the 1990s, the mid of second AI winter, the Internet started prevailing for commercial uses. I think one of the iconic events in this time was the source codes WWW (World Wide Web) were announced in 1993. Some of you might still remember that you little by little became able to transmit more data online in this time. That means people came to get more and more access to various datasets in those days, which is indispensable for machine learning tasks.

After all, we could not get HAL 9000 by the end of 2001, but instead we got Xbox console.

3, Video game industry and GPU

Even though research on neural networks stagnated in the 1990s the same period witnessed an advance in the computation of massive parallel linear transformations, due to their need in fields such as image processing.

Computer graphics move or rotate in 3d spaces, and that is also linear transformations. When you think about a car moving in a city, it is convenient to place the car, buildings, and other objects on a fixed 3d space. But when you need to make computer graphics of scenes of the city from a view point inside the car, you put a moving origin point in the car and see the city. The spatial information of the city is calculated as vectors from the moving origin point. Of course this is also linear transformations. Of course I am not talking about a dot or simple figures moving in the 3d spaces. Computer graphics are composed of numerous plane panels, and each of them have at least three vertexes, and they move on 3d spaces. Depending on viewpoints, you need project the 3d graphics in 3d spaces on 2d spaces to display the graphics on devices. You need to calculate which part of the panel is projected to which pixel on the display, and that is called rasterization. Plus, in order to get photophotorealistic image, you need to think about how lights from light sources reflect on the panel and projected on the display. And you also have to put some textures on groups of panels. You might also need to change color spaces, which is also linear transformations.

My point is, in short, you really need to do numerous linear transformations in parallel in image processing.

When it comes to the use of CGI in movies,  two pioneer movies were released during this time: Jurassic Park in 1993, and Toy Story in 1995. It is famous that Pixar used to be one of the departments in ILM (Industrial Light and Magic), founded by George Lucas, and Steve Jobs bought the department. Even though the members in Pixar had not even made a long feature film in their lives, after trial and errors, they made the first CGI animated feature movie. On the other hand, in order to acquire funds for the production of Schindler’s List (1993), Steven Spielberg took on Jurassic Park (1993), consequently changing the history of CGI through this “side job.”

Source: http://renderstory.com/jurassic-park-23-years-later/

*I think you have realized that George Lucas is mentioned almost everywhere in this article. His influences on technologies are not only limited to image processing, but also sound measuring system, nonlinear editing system. Photoshop was also originally developed under his company. I need another article series for this topic, but maybe not in Data Science Blog.

Source: https://editorial.rottentomatoes.com/article/5-technical-breakthroughs-in-star-wars-that-changed-movies-forever/

Considering that the first wire-frame computer graphics made and displayed by computers appeared in the scene of displaying the wire frame structure of Death Star in a war room, in Star Wars: A New Hope, the development of CGI was already astonishing at this time. But I think deep learning owe its development more to video game industry.

*I said that the Death Star scene is the first use of graphics made and DISPLAYED by computers, because I have to say one of the first graphics in movie MADE by computer dates back to the legendary title sequence of Vertigo(1958).

When it comes to 3D video games the processing unit has to constantly deal with real time commands from controllers. It is famous that GPU was originally specifically designed for plotting computer graphics. Video game market is the biggest in entertainment industry in general, and it is said that the quality of computer graphics have the strongest correlation with video games sales, therefore enhancing this quality is a priority for the video game console manufacturers.

One good example to see how much video games developed is comparing original Final Fantasy 7 and the remake one. The original one was released in 1997, the same year as when LSTM was invented. And recently  the remake version of Final Fantasy 7 was finally released this year. The original one was also made with very big budget, and it was divided into three CD-ROMs. The original one was also very revolutionary given that the former ones of Final Fantasy franchise were all 2d video retro style video games. But still the computer graphics looks like polygons, and in almost all scenes the camera angle was fixed in the original one. On the other hand the remake one is very photorealistic and you can move the angle of the camera as you want while you play the video game.

There were also fierce battles by graphic processor manufacturers in computer video game market in the 1990s, but personally I think the release of Xbox console was a turning point in the development of GPU. To be concrete, Microsoft adopted a type of NV20 GPU for Xbox consoles, and that left some room of programmability for developers. The chief architect of NV20, which was released under the brand of GeForce3, said making major changes in the company’s graphic chips was very risky. But that decision opened up possibilities of uses of GPU beyond computer graphics.

Source: https://de.wikipedia.org/wiki/Nvidia-GeForce-3-Serie

I think that the idea of a programmable GPU provided other scientific fields with more visible benefits after CUDA was launched. And GPU gained its position not only in deep learning, but also many other fields including making super computers.

*When it comes to deep learning, even GPUs have strong rivals. TPU(Tensor Processing Unit) made by Google, is specialized for deep learning tasks, and have astonishing processing speed. And FPGA(Field Programmable Gate Array), which was originally invented customizable electronic circuit, proved to be efficient for reducing electricity consumption of deep learning tasks.

*I am not so sure about this GPU part. Processing unit, including GPU is another big topic, that is beyond my capacity to be honest.  I would appreciate it if you could share your view and some references to confirm your opinion, on the comment section or via email.

*If you are interested you should see this video of game fans’ reactions to the announcement of Final Fantasy 7. This is the industry which grew behind the development of deep learning, and many fields where you need parallel computations owe themselves to the nerds who spent a lot of money for video games, including me.

*But ironically the engineers who invented the GPU said they did not play video games simply because they were busy. If you try to study the technologies behind video games, you would not have much time playing them. That is the reality.

We have seen that the in this second AI winter, Internet and GPU laid foundation of the next AI boom. But still the last piece of the puzzle is missing: let’s look at the breakthrough which solved the vanishing /exploding gradient problem of deep learning in the next section.

4, Pretraining of deep belief networks: “The Dawn of Deep Learning”

Some researchers say the invention of pretraining of deep belief network by Geoffrey Hinton was a breakthrough which put an end to the last AI winter. Deep belief networks are different type of networks from the neural networks we have discussed, but their architectures are similar to those of the neural networks. And it was also unknown how to train deep belief nets when they have several layers. Hinton discovered that training the networks layer by layer in advance can tackle vanishing gradient problems. And later it was discovered that you can do pretraining neural networks layer by layer with autoencoders.

*Deep belief network is beyond the scope of this article series. I have to talk about generative models, Boltzmann machine, and some other topics.

The pretraining techniques of neural networks is not mainstream anymore. But I think it is very meaningful to know that major deep learning techniques such as using ReLU activation functions, optimization with Adam, dropout, batch normalization, came up as more effective algorithms for deep learning after the advent of the pretraining techniques, and now we are in the third AI boom.

In the next next article we are finally going to work on LSTM. Specifically, I am going to offer a clearer guide to a well-made paper on LSTM, named “LSTM: A Search Space Odyssey.”

* I make study materials on machine learning, sponsored by DATANOMIQ. I do my best to make my content as straightforward but as precise as possible. I include all of my reference sources. If you notice any mistakes in my materials, including grammatical errors, please let me know (email: yasuto.tamura@datanomiq.de). And if you have any advice for making my materials more understandable to learners, I would appreciate hearing it.

[References]

[1] Taniguchi Tadahiro, “An Illustrated Guide to Artificial Intelligence”, (2010), Kodansha pp. 3-11
谷口忠大 著, 「イラストで学ぶ人工知能概論」, (2010), 講談社, pp. 3-11

[2] Francois Chollet, Deep Learning with Python,(2018), Manning , pp. 14-24

[3] Oketani Takayuki, “Machine Learning Professional Series: Deep Learning,” (2015), pp. 1-5, 151-156
岡谷貴之 著, 「機械学習プロフェッショナルシリーズ 深層学習」, (2015), pp. 1-5, 151-156

[4] Abigail See, Matthew Lamm, “Natural Language Processingwith Deep LearningCS224N/Ling284 Lecture 8:Machine Translation,Sequence-to-sequence and Attention,” (2020),
URL: http://web.stanford.edu/class/cs224n/slides/cs224n-2020-lecture08-nmt.pdf

[5]C. M. Bishop, “Pattern Recognition and Machine Learning,” (2006), Springer, pp. 192-196

[6] Daniel C. Dennett, “Cognitive Wheels: the Frame Problem of AI,” (1984), pp. 1-2

[7] Machiyama Tomohiro, “Understanding Cinemas of 1967-1979,” (2014), Yosensya, pp. 14-30
町山智浩 著, 「<映画の見方>が分かる本」,(2014), 洋泉社, pp. 14-30

[8] Harada Tatsuya, “Machine Learning Professional Series: Image Recognition,” (2017), pp. 156-157
原田達也 著, 「機械学習プロフェッショナルシリーズ 画像認識」, (2017), pp. 156-157

[9] Suyama Atsushi, “Machine Learning Professional Series: Bayesian Deep Learning,” (2019)岡谷貴之 須山敦志 著, 「機械学習プロフェッショナルシリーズ ベイズ深層学習」, (2019)

[10] “Understandable LSTM ~ With the Current Trends,” Qiita, (2015)
「わかるLSTM ~ 最近の動向と共に」, Qiita, (2015)
URL: https://qiita.com/t_Signull/items/21b82be280b46f467d1b

[11] Hisa Ando, “WEB+DB PRESS plus series: Technologies Supporting Processors – The World Endlessly Pursuing Speed,” (2017), Gijutsu-hyoron-sya, pp 313-317
Hisa Ando, 「WEB+DB PRESS plusシリーズ プロセッサを支える技術― 果てしなくスピードを追求する世界」, (2017), 技術評論社, pp. 313-317

[12] “Takahashi Yoshiki and Utamaru discuss George Lucas,” miyearnZZ Labo, (2016)
“高橋ヨシキと宇多丸 ジョージ・ルーカスを語る,” miyearnZZ Labo, (2016)
URL: https://miyearnzzlabo.com/archives/38865

[13] Katherine Bourzac, “Chip Hall of Fame: Nvidia NV20 The first configurable graphics processor opened the door to a machine-learning revolution,” IEEE SPECTRUM, (2018)
URL: https://spectrum.ieee.org/tech-history/silicon-revolution/chip-hall-of-fame-nvidia-nv20

Simple RNN

Simple RNN: the first foothold for understanding LSTM

*In this article “Densely Connected Layers” is written as “DCL,” and “Convolutional Neural Network” as “CNN.”

In the last article, I mentioned “When it comes to the structure of RNN, many study materials try to avoid showing that RNNs are also connections of neurons, as well as DCL or CNN.” Even if you manage to understand DCL and CNN, you can be suddenly left behind once you try to understand RNN because it looks like a different field. In the second section of this article, I am going to provide a some helps for more abstract understandings of DCL/CNN , which you need when you read most other study materials.

My explanation on this simple RNN is based on a chapter in a textbook published by Massachusetts Institute of Technology, which is also recommended in some deep learning courses of Stanford University.

First of all, you should keep it in mind that simple RNN are not useful in many cases, mainly because of vanishing/exploding gradient problem, which I am going to explain in the next article. LSTM is one major type of RNN used for tackling those problems. But without clear understanding forward/back propagation of RNN, I think many people would get stuck when they try to understand how LSTM works, especially during its back propagation stage. If you have tried climbing the mountain of understanding LSTM, but found yourself having to retreat back to the foot, I suggest that you read through this article on simple RNNs. It should help you to gain a solid foothold, and you would be ready for trying to climb the mountain again.

*This article is the second article of “A gentle introduction to the tiresome part of understanding RNN.”

1, A brief review on back propagation of DCL.

Simple RNNs are straightforward applications of DCL, but if you do not even have any ideas on DCL forward/back propagation, you will not be able to understand this article. If you more or less understand how back propagation of DCL works, you can skip this first section.

Deep learning is a part of machine learning. And most importantly, whether it is classical machine learning or deep learning, adjusting parameters is what machine learning is all about. Parameters mean elements of functions except for variants. For example when you get a very simple function f(x)=a + bx + cx^2 + dx^3, then x is a variant, and a, b, c, d are parameters. In case of classical machine learning algorithms, the number of those parameters are very limited because they were originally designed manually. Such functions for classical machine learning is useful for features found by humans, after trial and errors(feature engineering is a field of finding such effective features, manually). You adjust those parameters based on how different the outputs(estimated outcome of classification/regression) are from supervising vectors(the data prepared to show ideal answers).

In the last article I said neural networks are just mappings, whose inputs are vectors, matrices, or sequence data. In case of DCLs, inputs are vectors. Then what’s the number of parameters ? The answer depends on the the number of neurons and layers. In the example of DCL at the right side, the number of the connections of the neurons is the number of parameters(Would you like to try to count them? At least I would say “No.”). Unlike classical machine learning you no longer need to do feature engineering, but instead you need to design networks effective for each task and adjust a lot of parameters.

*I think the hype of AI comes from the fact that neural networks find features automatically. But the reality is difficulty of feature engineering was just replaced by difficulty of designing proper neural networks.

It is easy to imagine that you need an efficient way to adjust those parameters, and the method is called back propagation (or just backprop). As long as it is about DCL backprop, you can find a lot of well-made study materials on that, so I am not going to cover that topic precisely in this article series. Simply putting, during back propagation, in order to adjust parameters of a layer you need errors in the next layer. And in order calculate the errors of the next layer, you need errors in the next next layer.

*You should not think too much about what the “errors” exactly mean. Such “errors” are defined in this context, and you will see why you need them if you actually write down all the mathematical equations behind backprops of DCL.

The red arrows in the figure shows how errors of all the neurons in a layer propagate backward to a neuron in last layer. The figure shows only some sets of such errors propagating backward, but in practice you have to think about all the combinations of such red arrows in the whole back propagation(this link would give you some ideas on how DCLs work).

These points are minimum prerequisites for continuing reading this  RNN this article. But if you are planning to understand RNN forward/back propagation at  an abstract/mathematical level that you can read academic papers,  I highly recommend you to actually write down all the equations of DCL backprop. And if possible you should try to implement backprop of three-layer DCL.

2, Forward propagation of simple RNN

*For better understandings of the second and third section, I recommend you to download an animated PowerPoint slide which I prepared. It should help you understand simple RNNs.

In fact the simple RNN which we are going to look at in this article has only three layers. From now on imagine that inputs of RNN come from the bottom and outputs go up. But RNNs have to keep information of earlier times steps during upcoming several time steps because as I mentioned in the last article RNNs are used for sequence data, the order of whose elements is important. In order to do that, information of the neurons in the middle layer of RNN propagate forward to the middle layer itself. Therefore in one time step of forward propagation of RNN, the input at the time step propagates forward as normal DCL, and the RNN gives out an output at the time step. And information of one neuron in the middle layer propagate forward to the other neurons like yellow arrows in the figure. And the information in the next neuron propagate forward to the other neurons, and this process is repeated. This is called recurrent connections of RNN.

*To be exact we are just looking at a type of recurrent connections. For example Elman RNNs have simpler recurrent connections. And recurrent connections of LSTM are more complicated.

Whether it is a simple one or not, basically RNN repeats this process of getting an input at every time step, giving out an output, and making recurrent connections to the RNN itself. But you need to keep the values of activated neurons at every time step, so virtually you need to consider the same RNNs duplicated for several time steps like the figure below. This is the idea of unfolding RNN. Depending on contexts, the whole unfolded DCLs with recurrent connections is also called an RNN.

In many situations, RNNs are simplified as below. If you have read through this article until this point, I bet you gained some better understanding of RNNs, so you should little by little get used to this more abstract, blackboxed  way of showing RNN.

You have seen that you can unfold an RNN, per time step. From now on I am going to show the simple RNN in a simpler way,  based on the MIT textbook which I recomment. The figure below shows how RNN propagate forward during two time steps (t-1), (t).

The input \boldsymbol{x}^{(t-1)}at time step(t-1) propagate forward as a normal DCL, and gives out the output \hat{\boldsymbol{y}} ^{(t)} (The notation on the \boldsymbol{y} ^{(t)} is called “hat,” and it means that the value is an estimated value. Whatever machine learning tasks you work on, the outputs of the functions are just estimations of ideal outcomes. You need to adjust parameters for better estimations. You should always be careful whether it is an actual value or an estimated value in the context of machine learning or statistics). But the most important parts are the middle layers.

*To be exact I should have drawn the middle layers as connections of two layers of neurons like the figure at the right side. But I made my figure closer to the chart in the MIT textbook, and also most other study materials show the combinations of the two neurons before/after activation as one neuron.

\boldsymbol{a}^{(t)} is just linear summations of \boldsymbol{x}^{(t)} (If you do not know what “linear summations” mean, please scroll this page a bit), and \boldsymbol{h}^{(t)} is a combination of activated values of \boldsymbol{a}^{(t)} and linear summations of \boldsymbol{h}^{(t-1)} from the last time step, with recurrent connections. The values of \boldsymbol{h}^{(t)} propagate forward in two ways. One is normal DCL forward propagation to \hat{\boldsymbol{y}} ^{(t)} and \boldsymbol{o}^{(t)}, and the other is recurrent connections to \boldsymbol{h}^{(t+1)} .

These are equations for each step of forward propagation.

  • \boldsymbol{a}^{(t)} = \boldsymbol{b} + \boldsymbol{W} \cdot \boldsymbol{h}^{(t-1)} + \boldsymbol{U} \cdot \boldsymbol{x}^{(t)}
  • \boldsymbol{h}^{(t)}= g(\boldsymbol{a}^{(t)})
  • \boldsymbol{o}^{(t)} = \boldsymbol{c} + \boldsymbol{V} \cdot \boldsymbol{h}^{(t)}
  • \hat{\boldsymbol{y}} ^{(t)} = f(\boldsymbol{o}^{(t)})

*Please forgive me for adding some mathematical equations on this article even though I pledged not to in the first article. You can skip the them, but for some people it is on the contrary more confusing if there are no equations. In case you are allergic to mathematics, I prescribed some treatments below.

*Linear summation is a type of weighted summation of some elements. Concretely, when you have a vector \boldsymbol{x}=(x_0, x_1, x_2), and weights \boldsymbol{w}=(w_0,w_1, w_2), then \boldsymbol{w}^T \cdot \boldsymbol{x} = w_0 \cdot x_0 + w_1 \cdot x_1 +w_2 \cdot x_2 is a linear summation of \boldsymbol{x}, and its weights are \boldsymbol{w}.

*When you see a product of a matrix and a vector, for example a product of \boldsymbol{W} and \boldsymbol{v}, you should clearly make an image of connections between two layers of a neural network. You can also say each element of \boldsymbol{u}} is a linear summations all the elements of \boldsymbol{v}} , and \boldsymbol{W} gives the weights for the summations.

A very important point is that you share the same parameters, in this case \boldsymbol{\theta \in \{\boldsymbol{U}, \boldsymbol{W}, \boldsymbol{b}, \boldsymbol{V}, \boldsymbol{c} \}}, at every time step. 

And you are likely to see this RNN in this blackboxed form.

3, The steps of back propagation of simple RNN

In the last article, I said “I have to say backprop of RNN, especially LSTM (a useful and mainstream type or RNN), is a monster of chain rules.” I did my best to make my PowerPoint on LSTM backprop straightforward. But looking at it again, the LSTM backprop part still looks like an electronic circuit, and it requires some patience from you to understand it. If you want to understand LSTM at a more mathematical level, understanding the flow of simple RNN backprop is indispensable, so I would like you to be patient while understanding this step (and you have to be even more patient while understanding LSTM backprop).

This might be a matter of my literacy, but explanations on RNN backprop are very frustrating for me in the points below.

  • Most explanations just show how to calculate gradients at each time step.
  • Most study materials are visually very poor.
  • Most explanations just emphasize that “errors are back propagating through time,” using tons of arrows, but they lack concrete instructions on how actually you renew parameters with those errors.

If you can relate to the feelings I mentioned above, the instructions from now on could somewhat help you. And with the animated PowerPoint slide I prepared, you would have clear understandings on this topic at a more mathematical level.

Backprop of RNN , as long as you are thinking about simple RNNs, is not so different from that of DCLs. But you have to be careful about the meaning of errors in the context of RNN backprop. Back propagation through time (BPTT) is one of the major methods for RNN backprop, and I am sure most textbooks explain BPTT. But most study materials just emphasize that you need errors from all the time steps, and I think that is very misleading and confusing.

You need all the gradients to adjust parameters, but you do not necessarily need all the errors to calculate those gradients. Gradients in the context of machine learning mean partial derivatives of error functions (in this case J) with respect to certain parameters, and mathematically a gradient of J with respect to \boldsymbol{\theta \in \{\boldsymbol{U}, \boldsymbol{W}, \boldsymbol{b}^{(t)}, \boldsymbol{V}, \boldsymbol{c} \}}is denoted as ( \frac{\partial J}{\partial \boldsymbol{\theta}}  ). And another confusing point in many textbooks, including the MIT one, is that they give an impression that parameters depend on time steps. For example some study materials use notations like \frac{\partial J}{\partial \boldsymbol{\theta}^{(t)}}, and I think this gives an impression that this is a gradient with respect to the parameters at time step (t). In my opinion this gradient rather should be written as ( \frac{\partial J}{\partial \boldsymbol{\theta}} )^{(t)} . But many study materials denote gradients of those errors in the former way, so from now on let me use the notations which you can see in the figures in this article.

In order to calculate the gradient \frac{\partial J}{\partial \boldsymbol{x}^{(t)}} you need errors from time steps s (s \geq t) \quad (as you can see in the figure, in order to calculate a gradient in a colored frame, you need all the errors in the same color).

*To be exact, in the figure above I am supposed prepare much more arrows in \tau + 1 different colors  to show the whole process of RNN backprop, but that is not realistic. In the figure I displayed only the flows of errors necessary for calculating each gradient at time step 0, t, \tau.

*Another confusing point is that the \frac{\partial J}{\partial \boldsymbol{\ast ^{(t)}}}, \boldsymbol{\ast} \in \{\boldsymbol{a}^{(t)}, \boldsymbol{h}^{(t)}, \boldsymbol{o}^{(t)}, \dots \} are correct notations, because \boldsymbol{\ast} are values of neurons after forward propagation. They depend on time steps, and these are very values which I have been calling “errors.” That is why parameters do not depend on time steps, whereas errors depend on time steps.

As I mentioned before, you share the same parameters at every time step. Again, please do not assume that parameters are different from time step to time step. It is gradients/errors (you need errors to calculate gradients) which depend on time step. And after calculating errors at every time step, you can finally adjust parameters one time, and that’s why this is called “back propagation through time.” (It is easy to imagine that this method can be very inefficient. If the input is the whole text on a Wikipedia link, you need to input all the sentences in the Wikipedia text to renew parameters one time. To solve this problem there is a backprop method named “truncated BPTT,” with which you renew parameters based on a part of a text. )

And after calculating those gradients \frac{\partial J}{\partial \boldsymbol{\theta}^{(t)}} you can take a summation of them: \frac{\partial J}{\partial \boldsymbol{\theta}}=\sum_{t=0}^{t=\tau}{\frac{\partial J}{\partial \boldsymbol{\theta}^{(t)}}}. With this gradient \frac{\partial J}{\partial \boldsymbol{\theta}} , you can finally renew the value of \boldsymbol{\theta} one time.

At the beginning of this article I mentioned that simple RNNs are no longer for practical uses, and that comes from exploding/vanishing problem of RNN. This problem was one of the reasons for the AI winter which lasted for some 20 years. In the next article I am going to write about LSTM, a fancier type of RNN, in the context of a history of neural network history.

* I make study materials on machine learning, sponsored by DATANOMIQ. I do my best to make my content as straightforward but as precise as possible. I include all of my reference sources. If you notice any mistakes in my materials, including grammatical errors, please let me know (email: yasuto.tamura@datanomiq.de). And if you have any advice for making my materials more understandable to learners, I would appreciate hearing it.

Simple RNN

Prerequisites for understanding RNN at a more mathematical level

Writing the A gentle introduction to the tiresome part of understanding RNN Article Series on recurrent neural network (RNN) is nothing like a creative or ingenious idea. It is quite an ordinary topic. But still I am going to write my own new article on this ordinary topic because I have been frustrated by lack of sufficient explanations on RNN for slow learners like me.

I think many of readers of articles on this website at least know that RNN is a type of neural network used for AI tasks, such as time series prediction, machine translation, and voice recognition. But if you do not understand how RNNs work, especially during its back propagation, this blog series is for you.

After reading this articles series, I think you will be able to understand RNN in more mathematical and abstract ways. But in case some of the readers are allergic or intolerant to mathematics, I tried to use as little mathematics as possible.

Ideal prerequisite knowledge:

  • Some understanding on densely connected layers (or fully connected layers, multilayer perception) and how their forward/back propagation work.
  •  Some understanding on structure of Convolutional Neural Network.

*In this article “Densely Connected Layers” is written as “DCL,” and “Convolutional Neural Network” as “CNN.”

1, Difficulty of Understanding RNN

I bet a part of difficulty of understanding RNN comes from the variety of its structures. If you search “recurrent neural network” on Google Image or something, you will see what I mean. But that cannot be helped because RNN enables a variety of tasks.

Another major difficulty of understanding RNN is understanding its back propagation algorithm. I think some of you found it hard to understand chain rules in calculating back propagation of densely connected layers, where you have to make the most of linear algebra. And I have to say backprop of RNN, especially LSTM, is a monster of chain rules. I am planing to upload not only a blog post on RNN backprop, but also a presentation slides with animations to make it more understandable, in some external links.

In order to avoid such confusions, I am going to introduce a very simplified type of RNN, which I call a “simple RNN.” The RNN displayed as the head image of this article is a simple RNN.

2, How Neurons are Connected

How to connect neurons and how to activate them is what neural networks are all about. Structures of those neurons are easy to grasp as long as that is about DCL or CNN. But when it comes to the structure of RNN, many study materials try to avoid showing that RNNs are also connections of neurons, as well as DCL or CNN(*If you are not sure how neurons are connected in CNN, this link should be helpful. Draw a random digit in the square at the corner.). In fact the structure of RNN is also the same, and as long as it is a simple RNN, and it is not hard to visualize its structure.

Even though RNN is also connections of neurons, usually most RNN charts are simplified, using blackboxes. In case of simple RNN, most study material would display it as the chart below.

But that also cannot be helped because fancier RNN have more complicated connections of neurons, and there are no longer advantages of displaying RNN as connections of neurons, and you would need to understand RNN in more abstract way, I mean, as you see in most of textbooks.

I am going to explain details of simple RNN in the next article of this series.

3, Neural Networks as Mappings

If you still think that neural networks are something like magical spider webs or models of brain tissues, forget that. They are just ordinary mappings.

If you have been allergic to mathematics in your life, you might have never heard of the word “mapping.” If so, at least please keep it in mind that the equation y=f(x), which most people would have seen in compulsory education, is a part of mapping. If you get a value x, you get a value y corresponding to the x.

But in case of deep learning, x is a vector or a tensor, and it is denoted in bold like \boldsymbol{x} . If you have never studied linear algebra , imagine that a vector is a column of Excel data (only one column), a matrix is a sheet of Excel data (with some rows and columns), and a tensor is some sheets of Excel data (each sheet does not necessarily contain only one column.)

CNNs are mainly used for image processing, so their inputs are usually image data. Image data are in many cases (3, hight, width) tensors because usually an image has red, blue, green channels, and the image in each channel can be expressed as a height*width matrix (the “height” and the “width” are number of pixels, so they are discrete numbers).

The convolutional part of CNN (which I call “feature extraction part”) maps the tensors to a vector, and the last part is usually DCL, which works as classifier/regressor. At the end of the feature extraction part, you get a vector. I call it a “semantic vector” because the vector has information of “meaning” of the input image. In this link you can see maps of pictures plotted depending on the semantic vector. You can see that even if the pictures are not necessarily close pixelwise, they are close in terms of the “meanings” of the images.

In the example of a dog/cat classifier introduced by François Chollet, the developer of Keras, the CNN maps (3, 150, 150) tensors to 2-dimensional vectors, (1, 0) or (0, 1) for (dog, cat).

Wrapping up the points above, at least you should keep two points in mind: first, DCL is a classifier or a regressor, and CNN is a feature extractor used for image processing. And another important thing is, feature extraction parts of CNNs map images to vectors which are more related to the “meaning” of the image.

Importantly, I would like you to understand RNN this way. An RNN is also just a mapping.

*I recommend you to at least take a look at the beautiful pictures in this link. These pictures give you some insight into how CNN perceive images.

4, Problems of DCL and CNN, and needs for RNN

Taking an example of RNN task should be helpful for this topic. Probably machine translation is the most famous application of RNN, and it is also a good example of showing why DCL and CNN are not proper for some tasks. Its algorithms is out of the scope of this article series, but it would give you a good insight of some features of RNN. I prepared three sentences in German, English, and Japanese, which have the same meaning. Assume that each sentence is divided into some parts as shown below and that each vector corresponds to each part. In machine translation we want to convert a set of the vectors into another set of vectors.

Then let’s see why DCL and CNN are not proper for such task.

  • The input size is fixed: In case of the dog/cat classifier I have mentioned, even though the sizes of the input images varies, they were first molded into (3, 150, 150) tensors. But in machine translation, usually the length of the input is supposed to be flexible.
  • The order of inputs does not mater: In case of the dog/cat classifier the last section, even if the input is “cat,” “cat,” “dog” or “dog,” “cat,” “cat” there’s no difference. And in case of DCL, the network is symmetric, so even if you shuffle inputs, as long as you shuffle all of the input data in the same way, the DCL give out the same outcome . And if you have learned at least one foreign language, it is easy to imagine that the orders of vectors in sequence data matter in machine translation.

*It is said English language has phrase structure grammar, on the other hand Japanese language has dependency grammar. In English, the orders of words are important, but in Japanese as long as the particles and conjugations are correct, the orders of words are very flexible. In my impression, German grammar is between them. As long as you put the verb at the second position and the cases of the words are correct, the orders are also relatively flexible.

5, Sequence Data

We can say DCL and CNN are not useful when you want to process sequence data. Sequence data are a type of data which are lists of vectors. And importantly, the orders of the vectors matter. The number of vectors in sequence data is usually called time steps. A simple example of sequence data is meteorological data measured at a spot every ten minutes, for instance temperature, air pressure, wind velocity, humidity. In this case the data is recorded as 4-dimensional vector every ten minutes.

But this “time step” does not necessarily mean “time.” In case of natural language processing (including machine translation), which you I mentioned in the last section, the numberings of each vector denoting each part of sentences are “time steps.”

And RNNs are mappings from a sequence data to another sequence data.

In case of the machine translation above, the each sentence in German, English, and German is expressed as sequence data \boldsymbol{G}=(\boldsymbol{g}_1,\dots ,\boldsymbol{g}_{12}), \boldsymbol{E}=(\boldsymbol{e}_1,\dots ,\boldsymbol{e}_{11}), \boldsymbol{J}=(\boldsymbol{j}_1,\dots ,\boldsymbol{j}_{14}), and machine translation is nothing but mappings between these sequence data.

 

*At least I found a paper on the RNN’s capability of universal approximation on many-to-one RNN task. But I have not found any papers on universal approximation of many-to-many RNN tasks. Please let me know if you find any clue on whether such approximation is possible. I am desperate to know that. 

6, Types of RNN Tasks

RNN tasks can be classified into some types depending on the lengths of input/output sequences (the “length” means the times steps of input/output sequence data).

If you want to predict the temperature in 24 hours, based on several time series data points in the last 96 hours, the task is many-to-one. If you sample data every ten minutes, the input size is 96*6=574 (the input data is a list of 574 vectors), and the output size is 1 (which is a value of temperature). Another example of many-to-one task is sentiment classification. If you want to judge whether a post on SNS is positive or negative, the input size is very flexible (the length of the post varies.) But the output size is one, which is (1, 0) or (0, 1), which denotes (positive, negative).

*The charts in this section are simplified model of RNN used for each task. Please keep it in mind that they are not 100% correct, but I tried to make them as exact as possible compared to those in other study materials.

Music/text generation can be one-to-many tasks. If you give the first sound/word you can generate a phrase.

Next, let’s look at many-to-many tasks. Machine translation and voice recognition are likely to be major examples of many-to-many tasks, but here name entity recognition seems to be a proper choice. Name entity recognition is task of finding proper noun in a sentence . For example if you got two sentences “He said, ‘Teddy bears on sale!’ ” and ‘He said, “Teddy Roosevelt was a great president!” ‘ judging whether the “Teddy” is a proper noun or a normal noun is name entity recognition.

Machine translation and voice recognition, which are more popular, are also many-to-many tasks, but they use more sophisticated models. In case of machine translation, the inputs are sentences in the original language, and the outputs are sentences in another language. When it comes to voice recognition, the input is data of air pressure at several time steps, and the output is the recognized word or sentence. Again, these are out of the scope of this article but I would like to introduce the models briefly.

Machine translation uses a type of RNN named sequence-to-sequence model (which is often called seq2seq model). This model is also very important for other natural language processes tasks in general, such as text summarization. A seq2seq model is divided into the encoder part and the decoder part. The encoder gives out a hidden state vector and it used as the input of the decoder part. And decoder part generates texts, using the output of the last time step as the input of next time step.

Voice recognition is also a famous application of RNN, but it also needs a special type of RNN.

*To be honest, I don’t know what is the state-of-the-art voice recognition algorithm. The example in this article is a combination of RNN and a collapsing function made using Connectionist Temporal Classification (CTC). In this model, the output of RNN is much longer than the recorded words or sentences, so a collapsing function reduces the output into next output with normal length.

You might have noticed that RNNs in the charts above are connected in both directions. Depending on the RNN tasks you need such bidirectional RNNs.  I think it is also easy to imagine that such networks are necessary. Again, machine translation is a good example.

And interestingly, image captioning, which enables a computer to describe a picture, is one-to-many-task. As the output is a sentence, it is easy to imagine that the output is “many.” If it is a one-to-many task, the input is supposed to be a vector.

Where does the input come from? I mentioned that the last some layers in of CNN are closely connected to how CNNs extract meanings of pictures. Surprisingly such vectors, which I call a “semantic vectors” is the inputs of image captioning task (after some transformations, depending on the network models).

I think this articles includes major things you need to know as prerequisites when you want to understand RNN at more mathematical level. In the next article, I would like to explain the structure of a simple RNN, and how it forward propagate.

* I make study materials on machine learning, sponsored by DATANOMIQ. I do my best to make my content as straightforward but as precise as possible. I include all of my reference sources. If you notice any mistakes in my materials, please let me know (email: yasuto.tamura@datanomiq.de). And if you have any advice for making my materials more understandable to learners, I would appreciate hearing it.

As Businesses Struggle With ML, Automation Offers a Solution

In recent years, machine learning technology and the business solutions it enables has developed into a big business in and of itself. According to the industry analysts at IDC, spending on ML and AI technology is set to grow to almost $98 billion per year by 2023. In practical terms, that figure represents a business environment where ML technology has become a key priority for companies of every kind.

That doesn’t mean that the path to adopting ML technology is easy for businesses. Far from it. In fact, survey data seems to indicate that businesses are still struggling to get their machine learning efforts up and running. According to one such survey, it currently takes the average business as many as 90 days to deploy a single machine learning model. For 20% of businesses, that number is even higher.

From the data, it seems clear that something is missing in the methodologies that most companies rely on to make meaningful use of machine learning in their business workflows. A closer look at the situation reveals that the vast majority of data workers (analysts, data scientists, etc.) spend an inordinate amount of time on infrastructure work – and not on creating and refining machine learning models.

Streamlining the ML Adoption Process

To fix that problem, businesses need to turn to another growing area of technology: automation. By leveraging the latest in automation technology, it’s now possible to build an automated machine learning pipeline (AutoML pipeline) that cuts down on the repetitive tasks that slow down ML deployments and lets data workers get back to the work they were hired to do. With the right customized solution in place, a business’s ML team can:

  • Reduce the time spent on data collection, cleaning, and ingestion
  • Minimize human errors in the development of ML models
  • Decentralize the ML development process to create an ML-as-a-service model with increased accessibility for all business stakeholders

In short, an AutoML pipeline turns the high-effort functions of the ML development process into quick, self-adjusting steps handled exclusively by machines. In some use cases, an AutoML pipeline can even allow non-technical stakeholders to self-create ML solutions tailored to specific business use cases with no expert help required. In that way, it can cut ML costs, shorten deployment time, and allow data scientists to focus on tackling more complex modelling work to develop custom ML solutions that are still outside the scope of available automation techniques.

The Parts of an AutoML Pipeline

Although the frameworks and tools used to create an AutoML pipeline can vary, they all contain elements that conform to the following areas:

  • Data Preprocessing – Taking available business data from a variety of sources, cleaning it, standardizing it, and conducting missing value imputation
  • Feature Engineering – Identifying features in the raw data set to create hypotheses for the model to base predictions on
  • Model Selection – Choosing the right ML approach or hyperparameters to produce the desired predictions
  • Tuning Hyperparameters – Determining which hyperparameters help the model achieve optimal performance

As anyone familiar with ML development can tell you, the steps in the above process tend to represent the majority of the labour and time-intensive work that goes into creating a model that’s ready for real-world business use. It is also in those steps where the lion’s share of business ML budgets get consumed, and where most of the typical delays occur.

The Limitations and Considerations for Using AutoML

Given the scope of the work that can now become part of an AutoML pipeline, it’s tempting to imagine it as a panacea – something that will allow a business to reduce its reliance on data scientists going forward. Right now, though, the technology can’t do that. At this stage, AutoML technology is still best used as a tool to augment the productivity of business data teams, not to supplant them altogether.

To that end, there are some considerations that businesses using AutoML will need to keep in mind to make sure they get reliable, repeatable, and value-generating results, including:

  • Transparency – Businesses must establish proper vetting procedures to make sure they understand the models created by their AutoML pipeline, so they can explain why it’s making the choices or predictions it’s making. In some industries, such as in medicine or finance, this could even fall under relevant regulatory requirements.
  • Extensibility – Making sure the AutoML framework may be expanded and modified to suit changing business needs or to tackle new challenges as they arise.
  • Monitoring and Maintenance – Since today’s AutoML technology isn’t a set-it-and-forget-it proposition, it’s important to establish processes for the monitoring and maintenance of the deployment so it can continue to produce useful and reliable ML models.

The Bottom Line

As it stands today, the convergence of automation and machine learning holds the promise of delivering ML models at scale for businesses, which would greatly speed up the adoption of the technology and lower barriers to entry for those who have yet to embrace it. On the whole, that’s great news both for the businesses that will benefit from increased access to ML technology, as well as for the legions of data professionals tasked with making it all work.

It’s important to note, of course, that complete end-to-end ML automation with no human intervention is still a long way off. While businesses should absolutely explore building an automated machine learning pipeline to speed up development time in their data operations, they shouldn’t lose sight of the fact that they still need plenty of high-skilled data scientists and analysts on their teams. It’s those specialists that can make appropriate and productive use of the technology. Without them, an AutoML pipeline would accomplish little more than telling the business what it wants to hear.

The good news is that the AutoML tools that exist right now are sufficient to alleviate many of the real-world problems businesses face in their road to ML adoption. As they become more commonplace, there’s little doubt that the lead time to deploy machine learning models is going to shrink correspondingly – and that businesses will enjoy higher ROI and enhanced outcomes as a result.

Data Analytics & Artificial Intelligence Trends in 2020

Artificial intelligence has infiltrated all aspects of our lives and brought significant improvements.

Although the first thing that comes to most people’s minds when they think about AI are humanoid robots or intelligent machines from sci-fi flicks, this technology has had the most impressive advancements in the field of data science.

Big data analytics is what has already transformed the way we do business as it provides an unprecedented insight into a vast amount of unstructured, semi-structured, and structured data by analyzing, processing, and interpreting it.

Data and AI specialists and researchers are likely to have a field day in 2020, so here are some of the most important trends in this industry.

1. Predictive Analytics

As its name suggests, this trend will be all about using gargantuan data sets in order to predict outcomes and results.

This practice is slated to become one of the biggest trends in 2020 because it will help businesses improve their processes tremendously. It will find its place in optimizing customer support, pricing, supply chain, recruitment, and retail sales, to name just a few.

For example, Amazon has already been leveraging predictive analytics for its dynamic pricing model. Namely, the online retail giant uses this technology to analyze the demand for a particular product, competitors’ prices, and a number of other parameters in order to adjust its price.

According to stats, Amazon changes prices 2.5 million times a day so that a particular product’s cost fluctuates and changes every 10 minutes, which requires an extremely predictive analytics algorithm.

2. Improved Cybersecurity

In a world of advanced technologies where IoT and remotely controlled devices having top-notch protection is of critical importance.

Numerous businesses and individuals have fallen victim to ruthless criminals who can steal sensitive data or wipe out entire bank accounts. Even some big and powerful companies suffered huge financial and reputation blows due to cyber attacks they were subjected to.

This kind of crime is particularly harsh for small and medium businesses. Stats say that 60% of SMBs are forced to close down after being hit by such an attack.

AI again takes advantage of its immense potential for analyzing and processing data from different sources quickly and accurately. That’s why it’s capable of assisting cybersecurity specialists in predicting and preventing attacks.

In case that an attack emerges, the response time is significantly shorter, so that the worst-case scenario can be avoided.

When we’re talking about avoiding security risks, AI can improve enterprise risk management, too, by providing guidance and assisting risk management professionals.

3. Digital Workers

In 2020, an army of digital workers will transform the traditional workspace and take productivity to a whole new level.

Virtual assistants and chatbots are some examples of already existing digital workers, but it will be even more of them. According to research, this trend is one the rise, as it’s expected that AI software and robots will increase by 50% by 2022.

Robots will take over even some small tasks in the office. The point is to streamline the entire business process, and that can be achieved by training robots to perform small and simple tasks like human employees. The only difference will be that digital workers will do that faster and without any mistakes.

4. Hybrid Workforce

Many people worry that AI and automation will steal their jobs and render them unemployed.

Even the stats are bleak – AI will eliminate 1.8 million jobs. But, on the other hand, it will create 2.3 million new jobs.

So, our future is actually AI and humans working together, and that’s what will become the business normalcy in 2020.

Robotic process automation and different office digital workers will be in charge of tedious and repetitive tasks, while more sophisticated issues that require critical thinking and creativity will be human workers’ responsibility.

One of the most important things about creating this hybrid workforce is for businesses to openly discuss it with their employees and explain how these new technologies will be used. A regular workforce has to know that they will be working alongside machines whose job will be to speed up the processes and cut costs.

5. Process Intelligence

This AI trend will allow businesses to gain insight into their processes by using all the information contained in their system and creating an overall, real-time, and accurate visual model of all the processes.

What’s great about it is that it’s possible to see these processes from different perspectives – across departments, functions, staff, and locations.

With such a visual model, it’s possible to properly analyze these processes, identify potential bottlenecks, and eliminate them before they even begin to emerge.

Besides, as this is AI and data analytics at their best, this technology will also facilitate decision-making by predicting the future results of tech investments.

Needless to say, Process Intelligence will become an enterprise standard very soon, thanks to its ability to provide a better understanding and effective management of end-to-end processes.

As you can see, in 2020, these two advanced technologies will continue to evolve and transform the business landscape and change it for the better.

Interview – There is no stand-alone strategy for AI, it must be part of the company-wide strategy

Ronny FehlingRonny Fehling is Partner and Associate Director for Artificial Intelligence as the Boston Consulting Group GAMMA. With more than 20 years of continually progressive experience in leading business and technology innovation, spearheading digital transformation, and aligning the corporate strategy with Artificial Intelligence he industry-leading organizations to grow their top-line and kick-start their digital transformation.

Ronny Fehling is furthermore speaker of the Predictive Analytics World for Industry 4.0 in May 2020.

Data Science Blog: Mr. Fehling, you are consulting companies and business leaders about AI and how to get started with it. AI as a definition is often misleading. How do you define AI?

This is a good question. I think there are two ways to answer this:

From a technical definition, I often see expressions about “simulation of human intelligence” and “acting like a human”. I find using these terms more often misleading rather than helpful. I studied AI back when it wasn’t yet “cool” and still middle of the AI winter. And yes, we have much more compute power and access to data, but we also think about data in a very different way. For me, I typically distinguish between machine learning, which uses algorithms and statistical methods to identify patterns in data, and AI, which for me attempts to interpret the data in a given context. So machine learning can help me identify and analyze frequency patterns in text and even predict the next word I will type based on my history. AI will help me identify ‘what’ I’m writing about – even if I don’t explicitly name it. It can tell me that when I’m asking “I’m looking for a place to stay” that I might want to see a list of hotels around me. In other words: machine learning can detect correlations and similar patterns, AI uses machine learning to generate insights.

I always wondered why top executives are so frequently asking about the definition of AI because at first it seemed to me not as relevant to the discussion on how to align AI with their corporate strategy. However, I started to realize that their question is ultimately about “What is AI and what can it do for me?”.

For me, AI can do three things really good, which humans cannot really do and previous approaches couldn’t cope with:

  1. Finding similar patterns in historical data. Imagine 20 years of data like maintenance or repair documents of a manufacturing plant. Although they describe work done on a multitude of products due to a multitude of possible problems, AI can use this to look for a very similar situation based on a current problem description. This can be used to identify a common root cause as well as a common solution approach, saving valuable time for the operation.
  2. Finding correlations across time or processes. This is often used in predictive maintenance use cases. Here, the AI tries to see what similar events happen typically at some time before a failure happen. This way, it can alert the operator much earlier about an impending failure, say due to a change in the vibration pattern of the machine.
  3. Finding an optimal solution path based on many constraints. There are many problems in the business world, where choosing the optimal path based on complex situations is critical. Let’s say that suddenly a severe weather warning at an airport forces an airline to have to change their scheduling because of a reduced airport capacity. Delays for some aircraft can cause disruptions because passengers or personnel not being able to connect anymore. Knowing which aircraft to delay, which to cancel, which to switch while causing the minimal amount of disruption to passengers, crew, maintenance and ground-crew is something AI can help with.

The key now is to link these fundamental capabilities with the business context of the company and how it can ultimately help transform.

Data Science Blog: Companies are still starting with their own company-wide data strategy. And now they are talking about AI strategies. Is that something which should be handled separately?

In my experience – both based on having seen the implementations of several corporate data strategies as well as my upbringing at Oracle – the data strategy and AI strategy are co-dependent and cannot be separated. Very often I hear from clients that they think they first need to bring their data in order before doing AI project. And yes, without good data access, AI cannot really work. In fact, most of the time spent on AI is spent on processing, cleansing, understanding and contextualizing the data. However, you cannot really know what data will be needed in which form without knowing what you want to use it for. This is why strategies that handle data and AI separately mostly fail and generate huge costs.

Data Science Blog: What are the important steps for developing a good data strategy? Is there something like a general approach?

In my eyes, the AI strategy defines the data strategy step by step as more use cases are implemented. Rather than focusing too quickly at how to get all corporate data into a data lake, it will be much more important to start creating a use-case, technology and data governance. This governance has to be established once the AI strategy is starting to mature to enable the scale up and productization. At the beginning is to find the (very few) use-cases that can serve as light house projects to demonstrate (1) value impact, (2) a way to go from MVP to Pilot, and (3) how to address the data challenge. This will then more naturally identify the elements of governance, data access and technology that are required.

Data Science Blog: What are the most common questions from business leaders to you regarding AI? Why do they hesitate to get started?

By far it the most common question I get is: how do I get started? The hesitations often come from multiple sources like: “We don’t have the talent in house to do AI”, “Our data is not good enough”, “We don’t know which use-case to start with”, “It’s not easy for us to embrace agile and failure culture because our products are mission critical”, “We don’t know how much value this can bring us”.

Data Science Blog: Most managers prefer to start small and with lower risk. They seem to postpone bigger ideas to a later stage, at least some milestones should be reached. Is that a good idea or should they think bigger?

AI is often associated (rightfully so) with a new way of working – agile and embracing failures. Similarly, there is also the perception of significant cost to starting with AI (talent, technology, data). These perceptions often lead managers wanting to start with several smaller ambition use-cases where failure isn’t that grave. Once they have proven itself somehow, they would then move on to bigger projects. The problem with this strategy is on the one side that you fragment your few precious AI resources on too many projects and at the same time you cannot really demonstrate an impact since the projects weren’t chosen based on their impact potential.

The AI pioneers typically were successful by “thinking big, starting small and scaling fast”. You start by assessing the value potential of a use-case, for example: my current OEE (Overall Equipment Efficiency) is at 65%. There is an addressable loss of 25% which would grow my top line by $X. With the help of AI experts, you then create a hypothesis of how you think you can reduce that loss. This might be by choosing one specific equipment and 50% of the addressable loss. This is now the measure against which you define your failure or non-failure criteria. Once you have proven an MVP that can solve this loss, you scale up by piloting it in real-life setting and then scaling it to all the equipment. At every step of this process, you have a failure criterion that is measured by the impact value.


Virtual Edition, 11-12 MAY, 2020

The premier machine learning
conference for industry 4.0

This year Predictive Analytics World for Industry 4.0 runs alongside Deep Learning World and Predictive Analytics World for Healthcare.

Simple RNN

A gentle introduction to the tiresome part of understanding RNN

Just as a normal conversation in a random pub or bar in Berlin, people often ask me “Which language do you use?” I always answer “LaTeX and PowerPoint.”

I have been doing an internship at DATANOMIQ and trying to make straightforward but precise study materials on deep learning. I myself started learning machine learning in April of 2019, and I have been self-studying during this one-year-vacation of mine in Berlin.

Many study materials give good explanations on densely connected layers or convolutional neural networks (CNNs). But when it comes to back propagation of CNN and recurrent neural networks (RNNs), I think there’s much room for improvement to make the topic understandable to learners.

Many study materials avoid the points I want to understand, and that was as frustrating to me as listening to answers to questions in the Japanese Diet, or listening to speeches from the current Japanese minister of the environment. With the slightest common sense, you would always get the feeling “How?” after reading an RNN chapter in any book.

This blog series focuses on the introductory level of recurrent neural networks. By “introductory”, I mean prerequisites for a better and more mathematical understanding of RNN algorithms.

I am going to keep these posts as visual as possible, avoiding equations, but I am also going to attach some links to check more precise mathematical explanations.

This blog series is composed of five contents.:

  1. Prerequisites for understanding RNN at a more mathematical level
  2. Simple RNN: the first foothold for understanding LSTM
  3. A brief history of neural nets: everything you should know before learning LSTM
  4. Understanding LSTM forward propagation in two ways
  5. LSTM back propagation: following the flows of variables

 

Einführung in die Welt der Autoencoder

An wen ist der Artikel gerichtet?

In diesem Artikel wollen wir uns näher mit dem neuronalen Netz namens Autoencoder beschäftigen und wollen einen Einblick in die Grundprinzipien bekommen, die wir dann mit einem vereinfachten Programmierbeispiel festigen. Kenntnisse in Python, Tensorflow und neuronalen Netzen sind dabei sehr hilfreich.

Funktionsweise des Autoencoders

Ein Autoencoder ist ein neuronales Netz, welches versucht die Eingangsinformationen zu komprimieren und mit den reduzierten Informationen im Ausgang wieder korrekt nachzubilden.

Die Komprimierung und die Rekonstruktion der Eingangsinformationen laufen im Autoencoder nacheinander ab, weshalb wir das neuronale Netz auch in zwei Abschnitten betrachten können.

 

 

 

Der Encoder

Der Encoder oder auch Kodierer hat die Aufgabe, die Dimensionen der Eingangsinformationen zu reduzieren, man spricht auch von Dimensionsreduktion. Durch diese Reduktion werden die Informationen komprimiert und es werden nur die wichtigsten bzw. der Durchschnitt der Informationen weitergeleitet. Diese Methode hat wie viele andere Arten der Komprimierung auch einen Verlust.

In einem neuronalen Netz wird dies durch versteckte Schichten realisiert. Durch die Reduzierung von Knotenpunkten in den kommenden versteckten Schichten werden die Kodierung bewerkstelligt.

Der Decoder

Nachdem das Eingangssignal kodiert ist, kommt der Decoder bzw. Dekodierer zum Einsatz. Er hat die Aufgabe mit den komprimierten Informationen die ursprünglichen Daten zu rekonstruieren. Durch Fehlerrückführung werden die Gewichte des Netzes angepasst.

Ein bisschen Mathematik

Das Hauptziel des Autoencoders ist, dass das Ausgangssignal dem Eingangssignal gleicht, was bedeutet, dass wir eine Loss Funktion haben, die L(x , y) entspricht.

L(x, \hat{x})

Unser Eingang soll mit x gekennzeichnet werden. Unsere versteckte Schicht soll h sein. Damit hat unser Encoder folgenden Zusammenhang h = f(x).

Die Rekonstruktion im Decoder kann mit r = g(h) beschrieben werden. Bei unserem einfachen Autoencoder handelt es sich um ein Feed-Forward Netz ohne rückkoppelten Anteil und wird durch Backpropagation oder zu deutsch Fehlerrückführung optimiert.

Formelzeichen Bedeutung
\mathbf{x}, \hat{\mathbf{x}} Eingangs-, Ausgangssignal
\mathbf{W}, \hat{\mathbf{W}} Gewichte für En- und Decoder
\mathbf{B}, \hat{\mathbf{B}} Bias für En- und Decoder
\sigma, \hat{\sigma} Aktivierungsfunktion für En- und Decoder
L Verlustfunktion

Unsere versteckte Schicht soll mit \latex h gekennzeichnet werden. Damit besteht der Zusammenhang:

(1)   \begin{align*} \mathbf{h} &= f(\mathbf{x}) = \sigma(\mathbf{W}\mathbf{x} + \mathbf{B}) \\ \hat{\mathbf{x}} &= g(\mathbf{h}) = \hat{\sigma}(\hat{\mathbf{W}} \mathbf{h} + \hat{\mathbf{B}}) \\ \hat{\mathbf{x}} &= \hat{\sigma} \{ \hat{\mathbf{W}} \left[\sigma ( \mathbf{W}\mathbf{x} + \mathbf{B} )\right]  + \hat{\mathbf{B}} \}\\ \end{align*}

Für eine Optimierung mit der mittleren quadratischen Abweichung (MSE) könnte die Verlustfunktion wie folgt aussehen:

(2)   \begin{align*} L(\mathbf{x}, \hat{\mathbf{x}}) &= \mathbf{MSE}(\mathbf{x}, \hat{\mathbf{x}}) = \|  \mathbf{x} - \hat{\mathbf{x}} \| ^2 &=  \| \mathbf{x} - \hat{\sigma} \{ \hat{\mathbf{W}} \left[\sigma ( \mathbf{W}\mathbf{x} + \mathbf{B} )\right]  + \hat{\mathbf{B}} \} \| ^2 \end{align*}

 

Wir haben die Theorie und Mathematik eines Autoencoder in seiner Ursprungsform kennengelernt und wollen jetzt diese in einem (sehr) einfachen Beispiel anwenden, um zu schauen, ob der Autoencoder so funktioniert wie die Theorie es besagt.

Dazu nehmen wir einen One Hot (1 aus n) kodierten Datensatz, welcher die Zahlen von 0 bis 3 entspricht.

    \begin{align*} [1, 0, 0, 0] \ \widehat{=}  \ 0 \\ [0, 1, 0, 0] \ \widehat{=}  \ 1 \\ [0, 0, 1, 0] \ \widehat{=}  \ 2 \\ [0, 0, 0, 1] \ \widehat{=} \  3\\ \end{align*}

Diesen Datensatz könnte wie folgt kodiert werden:

    \begin{align*} [1, 0, 0, 0] \ \widehat{=}  \ 0 \ \widehat{=}  \ [0, 0] \\ [0, 1, 0, 0] \ \widehat{=}  \ 1 \ \widehat{=}  \  [0, 1] \\ [0, 0, 1, 0] \ \widehat{=}  \ 2 \ \widehat{=}  \ [1, 0] \\ [0, 0, 0, 1] \ \widehat{=} \  3 \ \widehat{=}  \ [1, 1] \\ \end{align*}

Damit hätten wir eine Dimensionsreduktion von vier auf zwei Merkmalen vorgenommen und genau diesen Vorgang wollen wir bei unserem Beispiel erreichen.

Programmierung eines einfachen Autoencoders

 

Typische Einsatzgebiete des Autoencoders sind neben der Dimensionsreduktion auch Bildaufarbeitung (z.B. Komprimierung, Entrauschen), Anomalie-Erkennung, Sequenz-to-Sequenz Analysen, etc.

Ausblick

Wir haben mit einem einfachen Beispiel die Funktionsweise des Autoencoders festigen können. Im nächsten Schritt wollen wir anhand realer Datensätze tiefer in gehen. Auch soll in kommenden Artikeln Variationen vom Autoencoder in verschiedenen Einsatzgebieten gezeigt werden.