Math Concepts for Supervised and Unsupervised Machine Learning

Supervised Learning

I. Introduction

• Definition of machine learning: Probability theory, linear algebra, calculus

• Types of machine learning: Supervised, unsupervised, reinforcement

• Applications of machine learning: Natural language processing, computer vision, fraud detection, recommendation systems

II. Supervised Learning A. Basics of Supervised Learning

• Types of problems:

  • Classification: Bayes' theorem, decision boundaries, softmax function, cross-entropy loss

  • Regression: Linear algebra (matrices, vectors), calculus (derivatives, partial derivatives), loss functions (mean squared error, mean absolute error)

• Learning process:

  • Training: Optimization algorithms (gradient descent, stochastic gradient descent, Adam), backpropagation (chain rule of derivatives)

  • Testing: Prediction, inference

• Evaluation metrics:

  • Classification: Confusion matrix, accuracy, precision, recall, F1 score, ROC curve, AUC

  • Regression: Mean squared error, mean absolute error, R-squared, explained variance

B. Linear Models

• Linear Regression:

  • Optimization algorithms: Normal equation, gradient descent

  • Linear algebra: Matrix multiplication, inverse, transpose

  • Calculus: Partial derivatives, gradients

• Logistic Regression:

  • Optimization algorithms: Gradient descent, Newton's method

  • Bayes' theorem: Probability theory, conditional probabilities

  • Linear algebra: Matrix multiplication

• Naive Bayes Classifier:

  • Bayes' theorem: Probability theory, conditional probabilities

  • Probability theory: Joint probabilities, marginal probabilities

• Support Vector Machines:

  • Optimization algorithms: Quadratic programming, dual problem, kernel methods

  • Calculus: Lagrange multipliers, partial derivatives

  • Linear algebra: Inner product, norms, Gram matrix

C. Tree-Based Models

• Decision Trees:

  • Entropy: Information theory, probability theory

  • Information gain: Entropy, conditional probabilities

  • Tree traversal: Depth-first search, breadth-first search

• Random Forests:

  • Ensemble learning: Bagging, bootstrap sampling

  • Decision trees: Gini impurity, information gain

• Gradient Boosted Trees:

  • Gradient descent: Optimization algorithm, partial derivatives

  • Decision trees: Regression trees, loss functions (mean squared error, mean absolute error)

D. Instance-Based Models

• k-Nearest Neighbors:

  • Distance metrics: Euclidean distance, Manhattan distance, cosine similarity

  • Voronoi diagrams: Geometry, Delaunay triangulation

E. Deep Learning Models

• Artificial Neural Networks:

  • Perceptron learning rule: Activation functions, linear combinations

  • Backpropagation: Chain rule of derivatives, gradients

  • Activation functions: Sigmoid, ReLU, softmax

• Convolutional Neural Networks:

  • Convolutional layers: Convolution operation, feature maps, stride, padding

  • Pooling layers: Max pooling, average pooling

  • ReLU activation: Nonlinearity, rectification

• Recurrent Neural Networks:

  • Backpropagation through time: Unfolding, gradients

  • LSTM units: Memory cells, input/output/forget gates, activation functions (sigmoid, tanh)

Unsupervised Learning

A. Basics of Unsupervised Learning

• Probability distributions and density estimation

• Clustering methods: k-means, hierarchical clustering, density-based clustering

• Dimensionality reduction techniques: principal component analysis (PCA), independent component analysis (ICA), non-negative matrix factorization (NMF), t-distributed stochastic neighbor embedding (t-SNE)

• Information theory: entropy, mutual information, KL divergence

• Optimization: gradient descent, stochastic gradient descent

B. Clustering Algorithms

• Distance measures: Euclidean distance, Manhattan distance, cosine similarity

• Objective functions: inertia, silhouette score

• Optimization: Lloyd's algorithm for k-means, agglomerative hierarchical clustering

C. Dimensionality Reduction Algorithms

• Linear algebra: eigenvectors, eigenvalues, singular value decomposition (SVD)

• Optimization: gradient descent, stochastic gradient descent, Adam optimization

• Information theory: entropy, mutual information, KL divergence

IV. Reinforcement Learning A. Basics of Reinforcement Learning

• Probability theory: Markov decision processes, stochastic policies, state transition probabilities

• Bellman equation: state-value function, action-value function, optimal policy

• Exploration vs. exploitation trade-off

B. Algorithms

• Q-Learning: Bellman equation, off-policy learning, epsilon-greedy policy

• Deep Q-Networks: experience replay, target network, neural network approximators

• Policy gradient methods: policy gradient theorem, REINFORCE algorithm

V. Advanced Topics A. Hyperparameter Tuning

• Optimization: grid search, random search, Bayesian optimization

• Cross-validation, overfitting

B. Regularization

• L1 regularization, L2 regularization, Elastic Net regularization

• Ridge regression, Lasso regression

C. Gradient Descent

• Batch gradient descent, stochastic gradient descent, mini-batch gradient descent

• Momentum, learning rate scheduling

D. Ensemble Methods

• Bagging: bootstrap aggregating, random forest

• Boosting: adaptive boosting, gradient boosting, XGBoost

• Stacking: meta-learner, ensemble of models

E. Transfer Learning

• Pretrained models, fine-tuning

• Domain adaptation, multi-task learning

F. Explainable AI

• Local interpretable model-agnostic explanations (LIME)

• Shapley Additive exPlanations (SHAP)