# Summary

M1, M2, M3,
P1,
Q1, Q2, Q3, Q4,
D1, D2,

# Lectures

Blank Slides: Part 1: PDF, Part 2: PDF, Part 3: PDF, Part 4: PDF,


Image by sandserifcomics via towards data science

# Other Materials

Lecture 1 Part 1 (Admin, 2021): Link and Link
Lecture 1 Part 2 (Supervised learning): Link
Lecture 1 Part 3 (Perceptron learning): Link
Lecture 2 Part 1 (Loss functions): Link
Lecture 2 Part 2 (Logistic regression): Link
Lecture 2 Part 3 (Convexity): Link

Lecture 3 Part 1 (Neural Network): Link
Lecture 3 Part 2 (Backpropogation): Link
Lecture 3 Part 3 (Multi-Layer Network): Link
Lecture 4 Part 1 (Stochastic Gradient): Link
Lecture 4 Part 2 (Multi-Class Classification): Link
Lecture 4 Part 3 (Regularization): 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

Neural Network: Link
Another Neural Network Demo: Link
Neural Network Videos by Grant Sanderson: Playlist
MNIST Neural Network Visualization: Link
Neural Network Simulator: Link
Overfitting: Link
Neural Network Snake: Video
Neural Network Car: Video
Neural Network Flappy Bird: Video
Neural Network Mario: Video
MyScript: algorithm Link demo Link
Maple Calculator: Link


Why does the (batch) perceptron algorithm work? Link
Why cannot use linear regression for binary classification? Link
How to use Perceptron update formula? Link
How to find the size of the hypothesis space for linear classifiers? Link (Part 1)

Why does gradient descent work? Link
Computation of Hessian of quadratic form Link
Computation of eigenvalues Link
Gradient descent for linear regression Link
What is the gradient descent step for cross-entropy loss with linear activation? Link (Part 1)
What is the sign of the next gradient descent step? Link (Part 2)
Which loss functions are equivalent to squared loss? Link (Part 3)
How to compute the gradient of the cross-entropy loss with linear activation? Link (Part 4)
How to find the location that minimizes the distance to multiple points? Link (Part 3)

How to derive logistic regression gradient descent step formula? Link
Gradient descent for logistic activation with squared error Link
How to compute logistic gradient descent? Link

How to construct XOR network? Link
How derive 2-layer neural network gradient descent step? Link
How derive multi-layer neural network gradient descent induction step? Link
How to find the missing weights in neural network given data set? Link
How many weights are used in one step of backpropogation? Link (Part 2)

Comparison between L1 and L2 regularization. Link
How to compute cross validation accuracy? 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.

# 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.

Neural network classifier for two layer network with logistic activation: \(\hat{y}_{i} = 1_{\left\{a^{\left(2\right)}_{i} \geq 0.5\right\}}\)
\(a^{\left(1\right)}_{ij} = \dfrac{1}{1 + \exp\left(- \left(\left(\displaystyle\sum_{j'=1}^{m} x_{ij'} w^{\left(1\right)}_{j'j}\right) + b^{\left(1\right)}_{j}\right)\right)}\), where \(m\) is the number of features (or input units), \(w^{\left(1\right)}_{j' j}\) is the layer \(1\) weight from input unit \(j'\) to hidden layer unit \(j\), \(b^{\left(1\right)}_{j}\) is the bias for hidden layer unit \(j\), \(a_{ij}^{\left(1\right)}\) is the layer \(1\) activation of instance \(i\) hidden unit \(j\).
\(a^{\left(2\right)}_{i} = \dfrac{1}{1 + \exp\left(- \left(\left(\displaystyle\sum_{j=1}^{h} a^{\left(1\right)}_{ij} w^{\left(2\right)}_{j}\right) + b^{\left(2\right)}\right)\right)}\), where \(h\) is the number of hidden units, \(w^{\left(2\right)}_{j}\) is the layer \(2\) weight from hidden layer unit \(j\), \(b^{\left(2\right)}\) is the bias for the output unit, \(a^{\left(2\right)}_{i}\) is the layer \(2\) activation of instance \(i\).
Stochastic gradient descent step for two layer network with squared loss and logistic activation:
\(w^{\left(1\right)}_{j' j} = w^{\left(1\right)}_{j' j} - \alpha \left(a^{\left(2\right)}_{i} - y_{i}\right) a^{\left(2\right)}_{i} \left(1 - a^{\left(2\right)}_{i}\right) w_{j}^{\left(2\right)} a_{ij}^{\left(1\right)} \left(1 - a_{ij}^{\left(1\right)}\right) x_{ij'}\).
\(b^{\left(1\right)}_{j} \leftarrow b^{\left(1\right)}_{j} - \alpha \left(a^{\left(2\right)}_{i} - y_{i}\right) a^{\left(2\right)}_{i} \left(1 - a^{\left(2\right)}_{i}\right) w_{j}^{\left(2\right)} a_{ij}^{\left(1\right)} \left(1 - a_{ij}^{\left(1\right)}\right)\).
\(w^{\left(2\right)}_{j} \leftarrow w^{\left(2\right)}_{j} - \alpha \left(a^{\left(2\right)}_{i} - y_{i}\right) a^{\left(2\right)}_{i} \left(1 - a^{\left(2\right)}_{i}\right) a_{ij}^{\left(1\right)}\).
\(b^{\left(2\right)} \leftarrow b^{\left(2\right)} - \alpha \left(a^{\left(2\right)}_{i} - y_{i}\right) a^{\left(2\right)}_{i} \left(1 - a^{\left(2\right)}_{i}\right)\).

Softmax activation for one layer networks: \(a_{ij} = \dfrac{\exp\left(- \left(w_{k^\top} x_{i} + b_{k}\right)\right)}{\displaystyle\sum_{k' = 1}^{K} \exp\left(- \left(w_{k'}^\top x_{i} + b_{k'}\right)\right)}\), where \(K\) is the number of classes (number of possible labels), \(a_{i k}\) is the activation of the output unit \(k\) for instance \(i\), \(y_{i k}\) is component \(k\) of the one-hot encoding of the label for instance \(i\).

L1 regularization (squared loss): \(\displaystyle\sum_{i=1}^{n} \left(a_{i} - y_{i}\right)^{2} + \lambda \left(\displaystyle\sum_{j=1}^{m} \left| w_{j} \right| + \left| b \right|\right)\), where \(\lambda\) is the regularization parameter.
L2 regularization (sqaured loss): \(\displaystyle\sum_{i=1}^{n} \left(a_{i} - y_{i}\right)^{2} + \lambda \left(\displaystyle\sum_{j=1}^{m} \left(w_{j}\right)^{2} + b^{2}\right)\).