Problem

This tutorial addresses the challenge of providing a comprehensive and self-contained understanding of regression analysis for students possessing only basic university-level mathematics. The goal is to bridge the gap between classical statistical modeling and modern machine learning practices, which often involve complex, nonlinear relationships in data that simple linear models cannot capture. A significant challenge in modeling is finding the right balance of model complexity to avoid both underfitting (where the model is too simple to capture patterns) and overfitting (where the model is too complex and learns noise, leading to poor generalization). While highly expressive models like deep learning offer powerful approximation capabilities, they frequently lack the interpretability of traditional parametric models, posing another challenge in understanding data-generating mechanisms.

Method

The paper systematically introduces a range of regression analysis models, starting with fundamental concepts and progressing to more advanced techniques. It covers linear regression for continuous outcomes, logistic regression for binary classification, and multinomial logistic (Softmax) regression for multi-class problems. The core methodology is built upon three essential elements: defining a suitable regression function, designing an appropriate loss function (e.g., Mean Squared Error for continuous labels and Cross-Entropy for discrete labels), and determining parameter estimation principles. For parameter estimation, both closed-form solutions (like Ordinary Least Squares for linear regression) and iterative optimization methods, particularly gradient descent and its variants (Batch, Stochastic, and Mini-Batch Gradient Descent), are detailed. The tutorial then extends to nonlinear regression through linear basis function models (including polynomial, Gaussian RBF, sigmoid, and Fourier bases) and neural networks, presenting deep learning as a form of nonlinear regression where basis functions are learnable. To combat overfitting, regularization techniques, specifically L2 (Ridge) and L1 (LASSO) regularization, are introduced and geometrically explained for their effects on model complexity and feature selection.

Results

The tutorial successfully demonstrates the construction and optimization of various regression models through detailed mathematical derivations and intuitive explanations. It shows that linear regression, logistic regression, and Softmax regression can handle different output types (continuous, binary, multi-class) by adapting their regression and loss functions. A key insight is the unified "prediction minus label times input" structure for gradients across these models, which is foundational to gradient-based optimization and the backpropagation algorithm for neural networks. The paper illustrates underfitting and overfitting using polynomial regression examples, showing how model complexity impacts training and generalization errors. Crucially, it demonstrates how regularization, particularly Ridge and LASSO, effectively controls model complexity, shrinks coefficients, and, in the case of LASSO, performs automatic feature selection by driving less important features' coefficients to zero. It also highlights the kernel trick as a method to implicitly compute inner products in high-dimensional feature spaces for linear basis function models.

Implications

This tutorial provides students and practitioners with a robust conceptual and technical foundation in regression analysis, bridging the gap between classical statistical principles and modern machine learning models, including deep learning. It underscores the importance of choosing appropriate model complexity, loss functions, and optimization algorithms for different data types and tasks. The detailed explanations of gradient descent and backpropagation demystify the training process for complex neural networks. By covering regularization techniques, the paper equips readers with essential tools to mitigate overfitting and enhance model generalization. Ultimately, the work promotes a deeper understanding of how regression models can be used for both prediction and interpretation, fostering informed decision-making in various intelligent computing applications and preparing students for further study in advanced artificial intelligence.