Perceptron Algorithm and Error Functions
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Questions and Answers

What is Error composed of?

  • classification error + margin error (correct)
  • only classification error
  • only margin error
  • neither classification error nor margin error

    • In the perceptron algorithm, how do we approach misclassified points?

  • by walking the line towards them (correct)
  • by removing them
  • by moving the line away from them
  • by ignoring them

    • What is the value of yi for points in the plus class?

  • 0
  • -1
  • Any value
  • 1 (correct)

    • What is the condition for data points in the plus class?

    <p>w.xi + b ≥ 1</p> Signup and view all the answers

    What can the perceptron algorithm be viewed as?

    <p>an algorithm that minimizes an error function</p> Signup and view all the answers

    What is the function of 𝒇 in Linear Classifiers?

    <p>to return the sign of (𝒘.𝒙 + 𝑏)</p> Signup and view all the answers

    What is the purpose of the margin in the Plus-Plane?

    <p>To maximize the distance between classes</p> Signup and view all the answers

    What does 𝒘 denote in Linear Classifiers?

    <p>the weight vector</p> Signup and view all the answers

    What is the equation for the decision boundary?

    <p>w.x + b = 0</p> Signup and view all the answers

    What is the sign of 𝒘.𝒙 + 𝑏 when the output is +1?

    <p>positive</p> Signup and view all the answers

    What is the role of the bias term b in the equation?

    <p>To shift the decision boundary</p> Signup and view all the answers

    What is the relationship between the Plus-Plane and the Minus-Plane?

    <p>They are parallel planes</p> Signup and view all the answers

    What is the purpose of the perceptron algorithm?

    <p>to classify data points correctly</p> Signup and view all the answers

    Where is the data classified when 𝒘.𝒙 + 𝑏 is positive?

    <p>as +1</p> Signup and view all the answers

    What is the condition for a point to be in the minus class?

    <p>xi + b ≤ -1</p> Signup and view all the answers

    What is the condition for data points in the minus class?

    <p>w.xi + b ≤ -1</p> Signup and view all the answers

    What is the value of b in the given equation?

    <p>b = 1</p> Signup and view all the answers

    What is the purpose of the decision boundary?

    <p>To separate the classes</p> Signup and view all the answers

    What is the formula for the margin width M?

    <p>M = 1/(2s)</p> Signup and view all the answers

    What is the equation for the Plus-plane?

    <p>w.x + b = 1</p> Signup and view all the answers

    What is the equation for the Minus-plane?

    <p>w.x + b = -1</p> Signup and view all the answers

    Why is the vector w perpendicular to the Plus Plane?

    <p>It is not stated in the provided text</p> Signup and view all the answers

    What is the relationship between xi and yi if yi = -1?

    <p>xi + b = -1</p> Signup and view all the answers

    What is the purpose of computing the margin width M?

    <p>It is not stated in the provided text</p> Signup and view all the answers

    What is the relationship between the vector w and the Plus Plane?

    <p>w is perpendicular to the Plus Plane</p> Signup and view all the answers

    What is the equation for the Plus Plane?

    <p>w.x + b = 1</p> Signup and view all the answers

    What is the equation for the Minus Plane?

    <p>w.x + b = -1</p> Signup and view all the answers

    What is the relationship between x+ and x-?

    <p>x+ is the closest Plus Plane point to x-</p> Signup and view all the answers

    What is the claim about the relationship between x+ and x-?

    <p>x+ = x- + l w for some value of l</p> Signup and view all the answers

    What is the margin width M?

    <p>M = 1 / (w.x + b)</p> Signup and view all the answers

    What is the equation for the point x+ in terms of x- and w?

    <p>x+ = x- + l w</p> Signup and view all the answers

    What is the value of w.(x - + l w) + b?

    <p>1</p> Signup and view all the answers

    What is the primary function of the C parameter in a Support Vector Machine?

    <p>To control the trade-off between the slack variable penalty and the width of the margin</p> Signup and view all the answers

    What is the effect of a small C parameter on the margin of a Support Vector Machine?

    <p>It leads to a large margin</p> Signup and view all the answers

    What is the purpose of the kernel trick in Support Vector Machines?

    <p>To map a higher-dimensional space to a lower-dimensional space</p> Signup and view all the answers

    What is the characteristic of a Radial Basis Function (RBF) kernel?

    <p>It is only defined by the relative position of the data points</p> Signup and view all the answers

    What is the role of λ in the cost function of a Support Vector Machine?

    <p>It controls the trade-off between the fit term and the regularization term</p> Signup and view all the answers

    What is the result of replacing the raw input variables with a much larger set of features in a Support Vector Machine?

    <p>A planar separator in the high-dimensional space</p> Signup and view all the answers

    What is the relationship between C and λ in a Support Vector Machine?

    <p>C is inversely proportional to λ</p> Signup and view all the answers

    What is the advantage of using the kernel trick in Support Vector Machines?

    <p>It allows for non-linear separators in the original feature space</p> Signup and view all the answers

    Study Notes

    Error and Perceptron Algorithm

    • Error = classification error + margin error
    • The perceptron algorithm minimizes an error function
    • The algorithm can be seen as an iterative process that adjusts a random line to correctly classify misclassified points

    Linear Classifiers

    • A linear classifier is defined as f(x, w, b) = sign(w.x + b)
    • w denotes the weights, x denotes the input, and b denotes the bias
    • The classifier outputs +1 or -1 depending on the sign of w.x + b

    Conditions for Optimal Separating Hyperplane

    • w.xi + b ≥ 1 if yi = 1 (points in plus class)
    • w.xi + b ≤ -1 if yi = -1 (points in minus class)

    Computing Margin Width

    • Margin width (M) can be computed using w and b
    • The vector w is perpendicular to the Plus Plane
    • The Plus-plane = {x : w.x + b = +1} and Minus-plane = {x : w.x + b = -1}
    • Margin width is the distance between the Plus-plane and Minus-plane

    Support Vector Machine – C Parameter

    • C is the regularization parameter that controls the trade-off between the slack variable penalty and width of the margin
    • Small C makes the constraints easy to ignore, leading to a large margin
    • Large C allows the constraints to be hard, leading to a small margin

    Linear Separability – Kernel Trick

    • Mapping input data to a higher dimension can make a non-linearly separable problem linearly separable
    • The kernel trick allows for efficient computation in high-dimensional spaces
    • Radial Basis Function (RBF) kernel is a type of kernel that only depends on the distance from a center point

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    Learn about the Perceptron algorithm, its connection to Neural Networks, and how it minimizes error functions to classify points correctly.

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