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The Universal Approximation Theorem is, very literally, the theoretical foundation of why neural networks work. Put simply, it states that a neural network with one hidden layer containing a sufficient but finite number of neurons can approximate any continuous function to a reasonable accuracy, under certain conditions for activation functions (namely, that they must be sigmoid-like).
Formulated in 1989 by George Cybenko only for sigmoid activations and proven by Kurt Hornik in 1991 to apply to all activation functions (the architecture of the neural network, not the choice of function, was the driver behind performance), its discovery was a significant driver in spurring the excited development of neural networks into the plethora of applications in which they are used today.
Most importantly, however, the theorem is an eye-opening explanation of why neural networks appear to behave so intelligently. Understanding it is a key step in developing a strong and deep understanding of neural networks.
Any continuous function on a compact (bounded, closed) set can be approximated by a piecewise function. Take, for instance, the sine wave between -3 and 3, which can be very convincingly approximated with three functions — two quadratic and one linear.
Graphed in Desmos.
Cybenko was more specific about this piecewise function, however, in that it could be constant, essentially consisting of several steps fitted to the function. With enough constant regions (‘steps’), one can reasonably estimate the function within the bounds it is given in.
Graphed in Desmos.
Based on this approximation, one could construct a network by delegating each neuron to one ‘step’. Using the weights and biases as ‘gates’ to determine which an input falls and hence which neuron should be activated, a neural network with a sufficient number of neurons could estimate a function simply be divvying it up into several constant regions.
For inputs that fall in a neuron’s delegated section, by blowing the weight to huge values, the final value approaches 1 (when evaluated using the sigmoid function). If it does not fall into the section, moving the weight towards negative infinity will yield a final result near 0. Using the sigmoid function as a “processor” of sorts to determine the degree of presence of a neuron, any function can be approximated almost perfectly, given an 6abundance of neurons. In multi-dimensional space, Cybenko generalized this idea, each neuron ‘controlling’ a hypercube of space within a multidimensional function.
The key point in the Universal Approximation Theorem is that instead of creating complex mathematical relationships between the input and output, it uses simple linear manipulations to divvy up the complicated function into many small, less complicated pieces, each of which are taken by one neuron.
Image created by Author.
Since Cybenko’s initial proof, many additional developments have been composed, such as testing out the Universal Approximation Theorem for different activation functions, like ReLU, which is unbounded (on one side), or with various architectures (recurrent, convolutional, etc.).
Regardless, all of these explorations center around one idea — neural networks find strength in numbers. Each neuron watches over one pattern or area of the feature space, whose size is determined by the number of neurons in the network. The less neurons there are, the more space each one will need to watch over, and hence approximation capability will go down. With more neurons, however, regardless of activation function, _any _function can be pieced together with many small fragments.
#machine-learning #ai #data-science #data-analysis #artificial-intelligence #data analysis
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Neural networks have been around for a long time, being developed in the 1960s as a way to simulate neural activity for the development of artificial intelligence systems. However, since then they have developed into a useful analytical tool often used in replace of, or in conjunction with, standard statistical models such as regression or classification as they can be used to predict or more a specific output. The main difference, and advantage, in this regard is that neural networks make no initial assumptions as to the form of the relationship or distribution that underlies the data, meaning they can be more flexible and capture non-standard and non-linear relationships between input and output variables, making them incredibly valuable in todays data rich environment.
In this sense, their use has took over the past decade or so, with the fall in costs and increase in ability of general computing power, the rise of large datasets allowing these models to be trained, and the development of frameworks such as TensforFlow and Keras that have allowed people with sufficient hardware (in some cases this is no longer even an requirement through cloud computing), the correct data and an understanding of a given coding language to implement them. This article therefore seeks to be provide a no code introduction to their architecture and how they work so that their implementation and benefits can be better understood.
Firstly, the way these models work is that there is an input layer, one or more hidden layers and an output layer, each of which are connected by layers of synaptic weights¹. The input layer (X) is used to take in scaled values of the input, usually within a standardised range of 0–1. The hidden layers (Z) are then used to define the relationship between the input and output using weights and activation functions. The output layer (Y) then transforms the results from the hidden layers into the predicted values, often also scaled to be within 0–1. The synaptic weights (W) connecting these layers are used in model training to determine the weights assigned to each input and prediction in order to get the best model fit. Visually, this is represented as:
#machine-learning #python #neural-networks #tensorflow #neural-network-algorithm #no code introduction to neural networks
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Talking about inspiration in the networking industry, nothing more than Autonomous Driving Network (ADN). You may hear about this and wondering what this is about, and does it have anything to do with autonomous driving vehicles? Your guess is right; the ADN concept is derived from or inspired by the rapid development of the autonomous driving car in recent years.
Driverless Car of the Future, the advertisement for “America’s Electric Light and Power Companies,” Saturday Evening Post, the 1950s.
The vision of autonomous driving has been around for more than 70 years. But engineers continuously make attempts to achieve the idea without too much success. The concept stayed as a fiction for a long time. In 2004, the US Defense Advanced Research Projects Administration (DARPA) organized the Grand Challenge for autonomous vehicles for teams to compete for the grand prize of $1 million. I remembered watching TV and saw those competing vehicles, behaved like driven by drunk man, had a really tough time to drive by itself. I thought that autonomous driving vision would still have a long way to go. To my surprise, the next year, 2005, Stanford University’s vehicles autonomously drove 131 miles in California’s Mojave desert without a scratch and took the $1 million Grand Challenge prize. How was that possible? Later I learned that the secret ingredient to make this possible was using the latest ML (Machine Learning) enabled AI (Artificial Intelligent ) technology.
Since then, AI technologies advanced rapidly and been implemented in all verticals. Around the 2016 time frame, the concept of Autonomous Driving Network started to emerge by combining AI and network to achieve network operational autonomy. The automation concept is nothing new in the networking industry; network operations are continually being automated here and there. But this time, ADN is beyond automating mundane tasks; it reaches a whole new level. With the help of AI technologies and other critical ingredients advancement like SDN (Software Defined Network), autonomous networking has a great chance from a vision to future reality.
In this article, we will examine some critical components of the ADN, current landscape, and factors that are important for ADN to be a success.
At the current stage, there are different terminologies to describe ADN vision by various organizations.
Even though slightly different terminologies, the industry is moving towards some common terms and consensus called autonomous networks, e.g. TMF, ETSI, ITU-T, GSMA. The core vision includes business and network aspects. The autonomous network delivers the “hyper-loop” from business requirements all the way to network and device layers.
On the network layer, it contains the below critical aspects:
On top of those, these capabilities need to be across multiple services, multiple domains, and the entire lifecycle(TMF, 2019).
No doubt, this is the most ambitious goal that the networking industry has ever aimed at. It has been described as the “end-state” and“ultimate goal” of networking evolution. This is not just a vision on PPT, the networking industry already on the move toward the goal.
David Wang, Huawei’s Executive Director of the Board and President of Products & Solutions, said in his 2018 Ultra-Broadband Forum(UBBF) keynote speech. (David W. 2018):
“In a fully connected and intelligent era, autonomous driving is becoming a reality. Industries like automotive, aerospace, and manufacturing are modernizing and renewing themselves by introducing autonomous technologies. However, the telecom sector is facing a major structural problem: Networks are growing year by year, but OPEX is growing faster than revenue. What’s more, it takes 100 times more effort for telecom operators to maintain their networks than OTT players. Therefore, it’s imperative that telecom operators build autonomous driving networks.”
Juniper CEO Rami Rahim said in his keynote at the company’s virtual AI event: (CRN, 2020)
“The goal now is a self-driving network. The call to action is to embrace the change. We can all benefit from putting more time into higher-layer activities, like keeping distributors out of the business. The future, I truly believe, is about getting the network out of the way. It is time for the infrastructure to take a back seat to the self-driving network.”
If you asked me this question 15 years ago, my answer would be “no chance” as I could not imagine an autonomous driving vehicle was possible then. But now, the vision is not far-fetch anymore not only because of ML/AI technology rapid advancement but other key building blocks are made significant progress, just name a few key building blocks:
#network-automation #autonomous-network #ai-in-network #self-driving-network #neural-networks
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Forward propagation is an important part of neural networks. Its not as hard as it sounds ;-)
This is part 2 in my series on neural networks. You are welcome to start at part 1 or skip to part 5 if you just want the code.
So, to perform gradient descent or cost optimisation, we need to write a cost function which performs:
In this article, we are dealing with (1) forward propagation.
In figure 1, we can see our network diagram with much of the details removed. We will focus on one unit in level 2 and one unit in level 3. This understanding can then be copied to all units. (ps. one unit is one of the circles below)
Our goal in forward prop is to calculate A1, Z2, A2, Z3 & A3
Just so we can visualise the X features, see figure 2 and for some more info on the data, see part 1.
As it turns out, this is quite an important topic for gradient descent. If you have not dealt with gradient descent, then check this article first. We can see above that we need 2 sets of weights. (signified by ø). We often still calls these weights theta and they mean the same thing.
We need one set of thetas for level 2 and a 2nd set for level 3. Each theta is a matrix and is size(L) * size(L-1). Thus for above:
Theta1 = 6x4 matrix
Theta2 = 7x7 matrix
We have to now guess at which initial thetas should be our starting point. Here, epsilon comes to the rescue and below is the matlab code to easily generate some random small numbers for our initial weights.
function weights = initializeWeights(inSize, outSize)
epsilon = 0.12;
weights = rand(outSize, 1 + inSize) * 2 * epsilon - epsilon;
end
After running above function with our sizes for each theta as mentioned above, we will get some good small random initial values as in figure 3
. For figure 1 above, the weights we mention would refer to rows 1 in below matrix’s.
Now, that we have our initial weights, we can go ahead and run gradient descent. However, this needs a cost function to help calculate the cost and gradients as it goes along. Before we can calculate the costs, we need to perform forward propagation to calculate our A1, Z2, A2, Z3 and A3 as per figure 1.
#machine-learning #machine-intelligence #neural-network-algorithm #neural-networks #networks
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Recurrent neural networks, also known as RNNs, are a class of neural networks that allow previous outputs to be used as inputs while having hidden states. RNN models are mostly used in the fields of natural language processing and speech recognition.
The vanishing and exploding gradient phenomena are often encountered in the context of RNNs. The reason why they happen is that it is difficult to capture long term dependencies because of multiplicative gradient that can be exponentially decreasing/increasing with respect to the number of layers.
Gated Recurrent Unit (GRU) and Long Short-Term Memory units (LSTM) deal with the vanishing gradient problem encountered by traditional RNNs, with LSTM being a generalization of GRU.
1D Convolution_ layer_ creates a convolution kernel that is convolved with the layer input over a single spatial (or temporal) dimension to produce a tensor of outputs. It is very effective for deriving features from a fixed-length segment of the overall dataset. A 1D CNN works well for natural language processing (NLP).
TensorFlow Datasets is a collection of datasets ready to use, with TensorFlow or other Python ML frameworks, such as Jax. All datasets are exposed as [_tf.data.Datasets_](https://www.tensorflow.org/api_docs/python/tf/data/Dataset)
, enabling easy-to-use and high-performance input pipelines.
“imdb_reviews”
This is a dataset for binary sentiment classification containing substantially more data than previous benchmark datasets. It provides a set of 25,000 highly polar movie reviews for training, and 25,000 for testing.
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
%matplotlib inline
import tensorflow as tf
import tensorflow_datasets
imdb, info=tensorflow_datasets.load("imdb_reviews", with_info=True, as_supervised=True)
imdb
info
train_data, test_data=imdb['train'], imdb['test']
training_sentences=[]
training_label=[]
testing_sentences=[]
testing_label=[]
for s,l in train_data:
training_sentences.append(str(s.numpy()))
training_label.append(l.numpy())
for s,l in test_data:
testing_sentences.append(str(s.numpy()))
testing_label.append(l.numpy())
training_label_final=np.array(training_label)
testing_label_final=np.array(testing_label)
vocab_size=10000
embedding_dim=16
max_length=120
trunc_type='post'
oov_tok='<oov>'
from tensorflow.keras.preprocessing.text import Tokenizer
from tensorflow.keras.preprocessing.sequence import pad_sequences
tokenizer= Tokenizer(num_words=vocab_size, oov_token=oov_tok)
tokenizer.fit_on_texts(training_sentences)
word_index=tokenizer.word_index
sequences=tokenizer.texts_to_sequences(training_sentences)
padded=pad_sequences(sequences, maxlen=max_length, truncating=trunc_type)
testing_sequences=tokenizer.texts_to_sequences(testing_sentences)
testing_padded=pad_sequences(testing_sequences, maxlen=max_length)
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense, Dropout, Embedding
#imdb #convolutional-network #long-short-term-memory #recurrent-neural-network #gated-recurrent-unit #neural networks
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The purpose of this project is to build and evaluate Recurrent Neural Networks(RNNs) for sentence-level classification tasks. I evaluate three architectures: a two-layer Long Short-Term Memory Network(LSTM), a two-layer Bidirectional Long Short-Term Memory Network(BiLSTM), and a two-layer BiLSTM with a word-level attention layer. Although they do learn useful vector representation, BiLSTM with attention mechanism focuses on necessary tokens when learning text representation. To that end, I’m using the 2019 Google Jigsaw published dataset on Kaggle labeled “Jigsaw Unintended Bias in Toxicity Classification.” The dataset includes 1,804,874 user comments, with the toxicity level being between 0 and 1. The final models can be used for filtering online posts and comments, social media policing, and user education.
RNNs are neural networks used for problems that require sequential data processing. For instance:
At each time step t of the input sequence, RNNs compute the output yt and an internal state update ht using the input xt and the previous hidden-state ht-1. They then pass information about the current time step of the network to the next. The hidden-state ht summarizes the task-relevant aspect of the past sequence of the input up to t, allowing for information to persist over time.
Recurrent Neural Network
Recurrent Neural Network
During training, RNNs re-use the same weight matrices at each time step. Parameter sharing enables the network to generalize to different sequence lengths. The total loss is a sum of all losses at each time step, the gradients with respect to the weights are the sum of the gradients at each time step, and the parameters are updated to minimize the loss function.
forward pass: compute the loss function
loss function
Backward Pass: compute the gradients
gradient equation
Although RNNs learn contextual representations of sequential data, they suffer from the exploding and vanishing gradient phenomena in long sequences. These problems occur due to the multiplicative gradient that can exponentially increase or decrease through time. RNNs commonly use three activation functions: RELU, Tanh, and Sigmoid. Because the gradient calculation also involves the gradient with respect to the non-linear activations, architectures that use a RELU activation can suffer from the exploding gradient problem. Architectures that use Tanh/Sigmoid can suffer from the vanishing gradient problem. Gradient clipping — limiting the gradient within a specific range — can be used to remedy the exploding gradient. However, for the vanishing gradient problem, a more complex recurrent unit with gates such as Gated Recurrent Unit (GRU) or Long Short-Term Memory (LSTM) can be used.
#ai #recurrent-neural-network #attention-network #machine-learning #neural-network