Machine Learning

Machine Learning is the process where a machine learns to identify features from an existing dataset so that it can accurately identify features in new data.

This definition isn’t strictly formal, and there’s no need for it to be. The term machine simply refers to a computer, or programming code. The dataset used for learning is known as a training set. If we compare machine learning to a student studying for exams, the analogy would be:

• Machine: Student
• Training Set: Past exam papers
• Features: Exam patterns
• New Data: Actual exam questions
• Learning: The process of solving past papers to prepare for actual exam questions

As of 2021, the most commonly used method for implementing machine learning is deep learning, which involves increasing the number of hidden layers in artificial neural networks. Artificial neural networks have significantly improved in performance recently and often deliver the best results. Before deep learning showed satisfactory performance, the mainstream of machine learning was models based on statistical theories.

To become proficient in machine learning, one must have knowledge in mathematics, statistics, and programming. Mathematical and statistical knowledge is required to understand the theories, and programming skills are necessary to implement these theories. In-depth study of machine learning theories requires not only knowledge of matrix and linear algebra but also measure theory, functional analysis, and more. Recently, research combining artificial intelligence with geometry, graph theory, partial differential equations, and other fields is also underway.¹

The following articles are written to be as accessible as possible for mathematics majors.

Basics

Learning Concepts

• Supervised and Unsupervised Learning
• Training/Validation/Test Sets
  • One-Hot Encoding
  • Data Augmentation
  • Commonly Used Datasets
    • MNIST Database
    • Iris Dataset
• Online Learning, Batch Learning, Mini-batch Learning

Optimization

• Loss Functions
• Gradient Descent, Stochastic Gradient Descent (SGD)
  • Momentum Method, Nesterov Accelerated Gradient (NAG)
  • Adaptive Learning Rates: AdaGrad, RMSProp, Adam
• 🔒 (07/24/17) Graduate Student Descent
  • 🔒 (07/13/24) Grid Search, Brute Force
• What is the Monte Carlo Method?

Classical Machine Learning

• Positive Definite Kernels and Reproducing Kernel Hilbert Space (k, H_k)
• Proof of the Representation Theorem

Linear Regression Models

• Linear Regression Models
  • Gradient Descent
  • Least Squares Method
  • Maximum Likelihood
• Support Vector Machines

Linear Classification Models

• Linear Classification Models
  • Least Squares Method
• Fisher’s Linear Discriminant
• Neyman-Pearson Criterion for Binary Classification
• Bayes Risk Classifier

Clustering

• k-Means Clustering
• Hierarchical Clustering

Sampling

• Monte Carlo Integration
• Rejection Sampling
• Importance Sampling
• Markov Chain Monte Carlo (MCMC)

Reinforcement Learning

• 🔒Basic Mathematics for Reinforcement Learning
• 🔒What is Reinforcement Learning
  • 🔒Value Functions
  • 🔒Exploration vs. Exploitation
• 🔒Multi-Armed Bandit Problem
  • 🔒Epsilon-Greedy Implementation with Julia
• 🔒Markov (Reward) Process
• 🔒Markov Decision Process

Deep Learning

Cheat Sheet: Equivalent Code in Flux, PyTorch, TensorFlow

Theory

• Weights
• Layers
  • Linear Layers
  • Convolutional Layers
  • Skip Connections
• Activation Functions
  • ReLU (ReLU(x) = \max{0, x})
  • Softplus (\zeta (x) = \ln (1 + e^x))
  • Sigmoid
  • Logistic (\sigma (x) = 1/(1+e^{-x}))
  • Softmax (\operatorname{softmax}(x) = )
• What is an Artificial Neural Network (ANN)?
• Definition of Perceptron
  • Perceptron Convergence Theorem
• What is Deep Learning?
  • Backpropagation Algorithm
• Mathematical Foundation of Deep Learning, Proof of the Universal Approximation Theorem
  • What is a Discriminant Function?
  • What is a Sigmoidal Function?
• Continual Learning in Deep Learning
• What is Computer Vision?
• Boltzmann Machine
  • Restricted Boltzmann Machine
  • Batch Learning Algorithm
  • Online Learning Algorithm
  • RBM for Classification
• Radial Basis Function

Regularization Techniques

• Overfitting and Regularization
• Dropout
• Paper Review: Do We Need Zero Training Loss After Achieving Zero Training Error?

Various Neural Networks

• [MLP (Multilayer Perceptron)]
• [CNN (Convolutional Neural Network)]
• PINN (Physics-Informed Neural Networks) Paper Review
• [U-net Paper Review]
  • [Implementation in Julia]

PyTorch

General

• Checking the Device a Model/Tensor is On .get_device()
• Random Sampling from a Given Distribution torch.distributions.Distribution().sample()
• Creating and Using Custom Datasets with Numpy Arrays TensorDataset, DataLoader
• Saving and Loading Weights, Models, Optimizers torch.save(model.state_dict())

Neural Networks

• Implementing a Multi-Layer Perceptron
• Defining Neural Networks with Lists and Loops nn.ModuleList
• Accessing Model Weights .weight.data, .bias.data
• Weight Initialization torch.nn.init

Tensors

• Modular Arithmetic fmod, remainder
• Handling Dimensions and Sizes .dim(), .ndim, .view(), .reshape(), .shape, .size()
• Creating Random Permutations and Shuffling Tensors torch.randperm, tensor[indices]
• Deep Copying .clone()
• Concatenating or Stacking torch.cat(), torch.stack()
• Padding torch.nn.functional.pad()
• Sorting torch.sort(), torch.argsort()

Errors

• Solving ‘RuntimeError: grad can be implicitly created only for scalar outputs’
• Fixing ‘TypeError: can’t convert cuda:0 device type tensor to numpy. Use Tensor.cpu() to copy the tensor to host memory first’ for Lists
• Addressing ‘RuntimeError: Boolean value of Tensor with more than one value is ambiguous’
• Resolving ‘RuntimeError: Parent directory does not exists’ When Saving Models

Julia

• Deep Learning Frameworks in Julia Flux.jl, Knet.jl, Lux.jl
• Using Machine Learning Datasets with MLDatasets.jl

Flux

• Handling Hidden Layers
• Implementing MLP and Optimizing with Gradient Descent
• One-Hot Encoding with onehot(), onebatch(), onecold()
• Implementing MLP and Approximating Non-linear Functions
• Implementing MLP and Training with MNIST
• 🔒(08/12/24) Using GPUs
• Setting Training and Testing Modes trainmode!, testmode!
• Getting Auto Differentiation of Neural Networks

References

• Christoper M. Bishop, Pattern Recognition annd Machine Learning (2006)
• Simon Haykin, Neural Networks and Learning Machines (3rd Edition, 2009)
• Trevor Hastie, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2nd Edition, 2017)
• 오일석, 기계 학습(MACHINE LEARNING) (2017)
• Richard S. Sutton, Reinforcement Learning: An Introduction (2nd Edition, 2018)