# Summary

M1, M2,
P1,
Q1,

# Lectures

Blank Slides: Part 1, Part 2,
Blank Slides (with blank pages for quiz questions): Part 1, Part 2,
Blank Slides with Quiz Questions: Part 1, Part 2,
Annotated Slides: Part 1, Part 2,


Image by sandserifcomics via towards data science
N/A

# Other Materials

Part 1 (Supervised learning): Link
Part 2 (Perceptron learning): Link
Part 3 (Loss functions): Link
Part 4 (Logistic regression): Link
Part 5 (Convexity): Link

Which face is real? Link
This X does not exist: Link
Turtle or Rifle: Link
Art or garbage game: Link
Guess two-thirds of the average? Link
Gradient Descent: Link
Optimization: Link
Neural Network: Link
Generative Adversarial Net: Link


Why does the (batch) perceptron algorithm work? Link
Why cannot use linear regression for binary classification? Link
Why does gradient descent work? Link
How to derive logistic regression gradient descent step formula? Link
Example (Quiz): Perceptron update formula Link
Example (Quiz): Gradient descent for logistic activation with squared error Link
Example (Quiz): Computation of Hessian of quadratic form Link
Example (Quiz): Computation of eigenvalues Link
Example (Homework): Gradient descent for linear regression Link
Video going through M1: Link

Checklist: Link, "math crib sheet": Link
Multivariate Calculus: Textbook, Chapter 16 and/or (Economics) Tutorials, Chapters 2 and 3.
Linear Algebra: Textbook, Chapters on Determinant and Eigenvalue.
Probability and Statistics: Textbook, Chapters 3, 4, 5.

# Perceptron Algorithm Demo

Data:

Weight:
Current Point:
Learning rate:
Fix b
Update

# Keywords and Notations

Training item: \(\left(x_{i}, y_{i}\right)\), where \(i \in \left\{1, 2, ..., n\right\}\) is the instance index, \(x_{ij}\) is the feature \(j\) of instance \(i\), \(j \in \left\{1, 2, ..., m\right\}\) is the feature index, \(x_{i} = \left(x_{i1}, x_{i2}, ...., x_{im}\right)\) is the feature vector of instance \(i\), and \(y_{i}\) is the true label of instance \(i\).
Test item: \(\left(x', y'\right)\), where \(j \in \left\{1, 2, ..., m\right\}\) is the feature index.

LTU Classifier: \(\hat{y}_{i} = 1_{\left\{w^\top x_{i} + b \geq 0\right\}}\), where \(w = \left(w_{1}, w_{2}, ..., w_{m}\right)\) is the weights, \(b\) is the bias, \(x_{i} = \left(x_{i1}, x_{i2}, ..., x_{im}\right)\) is the feature vector of instance \(i\), and \(\hat{y}_{i}\) is the predicted label of instance \(i\).
Perceptron algorithm update step: \(w = w - \alpha \left(a_{i} - y_{i}\right) x_{i}\), \(b = b - \alpha \left(a_{i} - y_{i}\right)\), \(a_{i} = 1_{\left\{w^\top x_{i} + b \geq 0\right\}}\), where \(a_{i}\) is the activation value of instance \(i\).

Zero-one loss minimization: \(\hat{f} = \mathop{\mathrm{argmin}}_{f \in \mathcal{H}} \displaystyle\sum_{i=1}^{n} 1_{\left\{f\left(x_{i}\right) \neq y_{i}\right\}}\), where \(\hat{f}\) is the optimal classifier, \(\mathcal{H}\) is the hypothesis space (set of functions to choose from).
Squared loss minimization of perceptrons: \(\left(\hat{w}, \hat{b}\right) = \mathop{\mathrm{argmin}}_{w, b} \dfrac{1}{2} \displaystyle\sum_{i=1}^{n} \left(a_{i} - y_{i}\right)^{2}\), \(a_{i} = g\left(w^\top x_{i} + b\right)\), where \(\hat{w}\) is the optimal weights, \(\hat{b}\) is the optimal bias, \(g\) is the activation function.

Logistic regression classifier: \(\hat{y}_{i} = 1_{\left\{a_{i} \geq 0.5\right\}}\), \(a_{i} = \dfrac{1}{1 + \exp\left(- \left(w^\top x_{i} + b\right)\right)}\).
Loss minimization problem: \(\left(\hat{w}, \hat{b}\right) = \mathop{\mathrm{argmin}}_{w, b} -\displaystyle\sum_{i=1}^{n} \left(y_{i} \log\left(a_{i}\right) + \left(1 - y_{i}\right) \log\left(1 - a_{i}\right)\right)\), \(a_{i} = \dfrac{1}{1 + \exp\left(- \left(w^\top x_{i} + b\right)\right)}\).
Batch gradient descrent step: \(w = w - \alpha \displaystyle\sum_{i=1}^{n} \left(a_{i} - y_{i}\right) x_{i}\), \(b = b - \alpha \displaystyle\sum_{i=1}^{n} \left(a_{i} - y_{i}\right)\), \(a_{i} = \dfrac{1}{1 + \exp\left(- \left(w^\top x_{i} + b\right)\right)}\), where \(\alpha\) is the learning rate.