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Questions and Answers
What is the Universal Approximation Theorem?
What is the Universal Approximation Theorem?
A feedforward network with a single layer is sufficient to represent any function.
Combining ReLUs gives a piecewise linear function.
Combining ReLUs gives a piecewise linear function.
True (A)
What type of classifiers are decision trees?
What type of classifiers are decision trees?
Non-linear classifiers
Which of the following statements about a single-layer network is true?
Which of the following statements about a single-layer network is true?
What is the distinction between discriminative and generative models?
What is the distinction between discriminative and generative models?
What roles do deep learning models like ChatGPT use?
What roles do deep learning models like ChatGPT use?
What percentage of data utilization trends can potentially save resources?
What percentage of data utilization trends can potentially save resources?
What has been learned regarding analytics in relation to our future on Earth?
What has been learned regarding analytics in relation to our future on Earth?
What is a perceptron?
What is a perceptron?
Which activation function can be used in perceptrons?
Which activation function can be used in perceptrons?
A perceptron can solve complex problems such as the XOR function on its own.
A perceptron can solve complex problems such as the XOR function on its own.
What does the activation function do in a perceptron?
What does the activation function do in a perceptron?
What is the role of the bias input in a perceptron?
What is the role of the bias input in a perceptron?
What are the building blocks of a neural network?
What are the building blocks of a neural network?
A perceptron computes its output $z$ using the formula $z = h(\sum_{i=0}^{D} w_i x_i)$, where $h$ is the ______.
A perceptron computes its output $z$ using the formula $z = h(\sum_{i=0}^{D} w_i x_i)$, where $h$ is the ______.
Linear regression produces a classifier function that can be considered a perceptron.
Linear regression produces a classifier function that can be considered a perceptron.
What limitation do perceptrons have?
What limitation do perceptrons have?
What mathematical operation does a perceptron perform?
What mathematical operation does a perceptron perform?
Neurons are inspired by ______ in biological systems.
Neurons are inspired by ______ in biological systems.
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Study Notes
Perceptrons
- A perceptron is a function that maps D-dimensional vectors to real numbers.
- It is a simple binary linear classifier, with the ability to compute the Boolean AND function.
- The perceptron model is inspired by the way neurons operate in the brain.
- A perceptron computes its output 𝑧 in two steps:
- Step 1: 𝑎 = 𝒘𝑇 𝒙 = σ𝐷 𝑖=0 𝑤𝑖 𝑥𝑖
- Step 2: 𝑧 = ℎ 𝑎
- The bias input (𝑥0) is always equal to 1.
- The bias weight (𝑤0) is optimised during training.
- The activation function (ℎ) used in a perceptron is either a step function or sigmoid function and can be changed based on the requirement.
- The sigmoidal function allows the use of gradient descent which is a powerful method for finding the optimal weights given a training dataset
- Examples of Boolean functions that can be computed by the perceptron are the AND, OR, and NOT function.
- The XOR function cannot be computed by a single perceptron but requires a neural network.
Neural Networks
- Built by using perceptrons as building blocks
- Inputs to some perceptrons are outputs of other perceptrons
- Can compute several functions
- Consists of units:
- input units
- perceptrons
- Each unit is connected to another unit by weights denoted by 𝑤𝑗𝑖.
- Weights are optimised by a learning method
- Neural networks are organized into layers:
- input layer
- output layer
- hidden layers (can be zero or multiple in between input and output layer)
- The outputs of the hidden layers serve as inputs to the next layer. This process continues until the output layer is reached.
- The XOR function can be computed by a neural network consisting of three units with two inputs (𝑥1, 𝑥2) and one output.
- The output of the XOR function can be represented by the following function: A OR B (unit 3) A AND (NOT B) (unit 5) A AND B (unit 4)
Universal Approximation Theorem
- Provides a theoretical foundation for the power of neural networks
- A feedforward network with a single layer is sufficient to represent any continuous function
- Uses rectified linear units (ReLUs)
- This theorem highlights that neural networks are capable of approximating any function, given sufficient complexity (neurons, layers).
- The theorem states: "A feedforward network with a single layer is sufficient to represent any function"
- This theorem allows us to represent a decision tree with a single layer by using ReLUs, which are piecewise linear functions.
- In simpler terms, ReLUs, can can take on a value of zero for any input less than zero and the input value itself if the input is greater than or equal to zero.
- These ReLU functions can be used to approximate any function by gradually increasing the number of ReLUs (Neurons in a single layer) by creating a piecewise linear function
The Importance of Activation Functions
- Activation functions are crucial for introducing non-linearity into neural networks.
- If we didn't have activation functions, neural networks would just be linear transformations, limiting their representational power.
- Examples of activation functions: step, sigmoid, ReLU
- The chosen activation function can massively impact the network's learning and performance.
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