Predicting the Future: A Guide to Regression Analysis

Author: Denis Avetisyan

From simple linear models to the power of deep learning, this review offers a complete exploration of techniques for understanding relationships in data.

Rental prices in the vicinity of a university are subject to regression analysis, demonstrating quantifiable relationships between property characteristics and market value.

This article provides a comprehensive overview of regression analysis, covering linear and non-linear models, gradient descent, regularization, and extensions to logistic and softmax regression.

Despite the increasing complexity of modern datasets, regression analysis remains a foundational technique for uncovering relationships between variables. This document, ‘A Tutorial on Regression Analysis: From Linear Models to Deep Learning — Lecture Notes on Artificial Intelligence’, provides a self-contained exploration of the topic, bridging classical statistical modeling with contemporary machine learning approaches. It systematically details core concepts-from ordinary least squares and gradient descent to regularization and logistic regression-equipping readers with the tools to build and optimize predictive models. Ultimately, how can a firm grasp of these regression fundamentals empower further innovation in artificial intelligence and data science?

The Essence of Relationship Modeling

Regression analysis serves as a powerful tool for understanding and quantifying the associations between variables, particularly when the outcome of interest is a continuous value. Rather than simply identifying if a relationship exists, regression delves into the nature of that relationship, attempting to model how changes in one or more predictor variables correspond to changes in the outcome. This predictive capability extends beyond mere description; it allows for estimations of future outcomes based on known values of the predictors. For instance, regression can be used to predict a student’s test score ($y$) based on the number of hours studied ($x$), or to forecast sales ($y$) based on advertising expenditure ($x$). The strength and direction of these relationships are mathematically defined, forming the basis for informed decision-making and accurate forecasting across diverse fields.

The regression function serves as the bedrock of predictive modeling, mathematically formalizing the link between variables. It isn’t simply a statement of correlation, but a precise equation – often expressed as $y = f(x) + \epsilon$ – that estimates the expected value of a dependent variable, y, given the value of one or more independent variables, x. Here, $f(x)$ represents the deterministic component of the relationship-the core prediction-while $\epsilon$ accounts for the inherent randomness and unmeasured factors contributing to variation. Establishing this function requires statistical techniques to best approximate the true underlying relationship from observed data, allowing for not just description of past trends, but also informed predictions about future outcomes. This mathematical definition provides a powerful framework for understanding and quantifying how changes in one variable systematically influence another, making it indispensable across diverse fields like economics, healthcare, and engineering.

The Softmax regression model utilizes a probabilistic approach to assign input data to multiple classes.

The Pursuit of Optimal Parameters

Parameter estimation is the process of identifying the values for the parameters within a regression function that yield the most accurate predictions of a dependent variable. The objective is to minimize the difference between predicted values and observed data points; this is commonly quantified using a loss function. A regression function, denoted as $f(x; \theta)$, maps input variables $x$ to a predicted output, where $\theta$ represents the set of parameters being estimated. The quality of the estimated parameters directly impacts the model’s predictive performance and its ability to generalize to unseen data. Different estimation techniques, such as Ordinary Least Squares and Gradient Descent, are employed to find the optimal parameter values that minimize the chosen loss function.

Ordinary Least Squares (OLS) is a statistical method used to estimate the parameters of a linear regression model by minimizing the sum of the squared differences between observed and predicted values. Mathematically, given a set of observations $(x_i, y_i)$ for $i = 1, …, n$, and a linear model $y = \beta_0 + \beta_1x$, OLS aims to find the values of $\beta_0$ and $\beta_1$ that minimize the Residual Sum of Squares (RSS), defined as $RSS = \sum_{i=1}^{n} (y_i – \hat{y}_i)^2$, where $\hat{y}_i$ is the predicted value for $x_i$. The resulting estimators for $\beta_0$ and $\beta_1$ possess desirable statistical properties, including being unbiased and having minimum variance under certain assumptions regarding the error terms.

Gradient Descent is an iterative optimization algorithm used to find the minimum of a function, specifically the ‘Loss Function’ in the context of parameter estimation. The algorithm operates by repeatedly adjusting model parameters in the direction of the steepest descent of the Loss Function, calculated using the gradient – a vector of partial derivatives. The size of each adjustment is determined by the ‘learning rate’, a hyperparameter controlling the step size. The process continues until a minimum is reached – either a local minimum or, ideally, the global minimum – or until the change in the Loss Function falls below a predefined threshold. Mathematically, the parameter update rule is expressed as $parameter = parameter – learning\_rate * gradient$.

The curvature of a multivariate function, as defined by its Hessian, determines the nature of its extrema, resulting in bowl-shaped local minima, dome-shaped local maxima, or saddle-shaped saddle points.

The Efficiency of Descent: A Comparative View

Batch Gradient Descent calculates the gradient of the cost function using the entire training dataset in each iteration of the optimization process. While this approach guarantees a more stable and accurate gradient estimate, its computational cost scales linearly with the dataset size, denoted as $O(n)$, where $n$ is the number of data points. Consequently, each iteration becomes increasingly time-consuming as the dataset grows, making it impractical for large-scale machine learning problems. The need to process the entire dataset before updating the model’s parameters limits the algorithm’s efficiency and scalability, particularly in scenarios involving millions or billions of data instances.

Stochastic Gradient Descent (SGD) achieves faster iteration times compared to Batch Gradient Descent by updating model parameters after evaluating each individual data point. This contrasts with methods that require processing the entire dataset before each update. While this accelerates learning, the use of a single data point introduces significant noise into the gradient estimation. This noise results in a less stable convergence path, potentially causing oscillations and preventing precise attainment of the global minimum. The magnitude of the noise is inversely proportional to the batch size; however, smaller batch sizes, while accelerating learning, require careful tuning of the learning rate to avoid divergence or suboptimal solutions. The update rule in SGD can be expressed as $w = w – \eta \nabla J(w; x^{(i)}, y^{(i)})$ where $\eta$ is the learning rate, and $(x^{(i)}, y^{(i)})$ represents the $i$-th data point.

Mini-batch gradient descent improves upon both batch and stochastic gradient descent by calculating the gradient using a randomly selected subset of the training data, known as a mini-batch. Typical mini-batch sizes range from 32 to 512, though optimal size depends on the specific dataset and model. This approach reduces the variance introduced by stochastic gradient descent, leading to more stable convergence, while simultaneously decreasing the computational cost per iteration compared to batch gradient descent. The resulting gradient is an approximation of the true gradient calculated on the entire dataset, but the averaged gradient over multiple mini-batches provides a robust and efficient path towards minimizing the loss function. This makes mini-batch gradient descent the most commonly used optimization algorithm in practice, particularly for training large-scale machine learning models.

Batch gradient descent, stochastic gradient descent, and mini-batch gradient descent each exhibit unique convergence paths, as illustrated in the figure sourced from NgMiniBatchGD.

The Perils of Adaptation: Overfitting and Underfitting

Overfitting represents a significant challenge in model building, arising when a predictive model learns the training data too well, including its random fluctuations and noise. Instead of identifying the underlying relationships, the model essentially memorizes the training set. This leads to excellent performance on the data it was trained on, but a marked inability to accurately predict outcomes for new, unseen data. Imagine a student who memorizes answers to practice questions instead of understanding the concepts; they will perform well on the practice test but struggle with novel problems. The result is a model with poor generalization capability, highlighting the importance of building models that capture true patterns rather than simply replicating the specifics of the training data.

Regularization offers a crucial strategy for building predictive models that generalize well beyond the training data. It operates by modifying the standard loss function – the metric a model aims to minimize – with a penalty term. This penalty discourages overly complex models, those with extremely large coefficients, which are prone to memorizing the training data rather than learning the underlying relationships. By adding this constraint, the model is nudged towards simplicity, prioritizing a balance between fitting the observed data and maintaining a compact representation. The strength of this penalty is controlled by a hyperparameter, allowing practitioners to fine-tune the model’s complexity and prevent it from becoming overly sensitive to noise, ultimately leading to improved performance on unseen data.

To combat overfitting, regularization techniques such as Ridge and LASSO regression offer distinct approaches to model complexity. Ridge regression, employing L2 regularization, minimizes the impact of each feature by shrinking their corresponding coefficients towards zero, but rarely eliminating them entirely. This encourages a balanced contribution from all predictors. Conversely, LASSO regression utilizes L1 regularization, which not only shrinks coefficients but also forces some to become precisely zero, effectively performing feature selection and simplifying the model by discarding irrelevant variables. The choice between the two depends on the dataset; Ridge is effective when most features contribute to the outcome, while LASSO excels when only a subset of features are truly important, resulting in a more interpretable and potentially more accurate model by focusing on the most salient predictors.

Underfitting represents a fundamental challenge in model building, arising when a chosen model lacks the capacity to discern the intricate relationships within the data. This simplification results in poor performance, not because the model has memorized noise, but because it fundamentally fails to capture the signal. The consequences manifest as both high bias and low variance; the model consistently makes inaccurate predictions, and its performance doesn’t substantially change with different training datasets. Addressing underfitting typically involves increasing model complexity – perhaps by adding more layers to a neural network or utilizing higher-order polynomial features in a regression model. Alternatively, feature engineering can prove vital, transforming existing data into representations that more clearly reveal the underlying patterns, allowing even a simpler model to achieve better results. Ultimately, overcoming underfitting requires a careful balance between model complexity and the richness of the features used to represent the data.

Regularization techniques effectively prevent overfitting by constraining model complexity.

Beyond Linearity: Expanding Model Capacity

Polynomial regression builds upon the foundation of linear regression by introducing polynomial terms – such as $x^2$ or $x^3$ – to the model. This seemingly simple addition dramatically expands the model’s capacity to represent data exhibiting non-linear relationships. While standard linear regression seeks a straight-line fit, polynomial regression allows the model to curve and adapt to more complex patterns. By incorporating these higher-order terms, the model can capture interactions and nuances within the data that would otherwise be missed, leading to a more accurate and insightful representation of the underlying phenomenon. This flexibility is particularly valuable when dealing with datasets where the relationship between variables isn’t a simple straight line, but rather a curve or a more intricate shape.

Linear basis function models represent a powerful generalization of standard linear regression by introducing non-linear transformations to the input features before applying the linear model. Instead of directly using the original input values, these models apply functions – such as polynomials, exponentials, or trigonometric functions – to each feature, effectively creating a new set of features that can capture more complex relationships. This transformation allows the model to fit data that exhibits non-linear patterns without abandoning the computational efficiency and interpretability of a linear model, as the final prediction is still a linear combination of these transformed features. By strategically choosing basis functions, the model gains the flexibility to approximate a wider range of functions and better represent the underlying data generating process, leading to improved predictive accuracy in scenarios where simple linear relationships are insufficient.

The ability to model complex relationships is paramount in numerous real-world applications, as many phenomena defy simple linear explanations. Consider, for example, the trajectory of a projectile – while a linear model might approximate its path over a very short distance, accurately predicting its arc requires accounting for the quadratic effect of gravity. Similarly, in epidemiology, the spread of infectious diseases often exhibits exponential growth initially, followed by a leveling-off as immunity develops, a pattern impossible to capture with a linear regression. Even in economic forecasting, consumer behavior and market dynamics frequently involve non-linear interactions, necessitating models capable of representing these intricacies. Therefore, extending beyond linear models isn’t merely a mathematical refinement; it’s a fundamental requirement for building predictive tools that reflect the true complexity of the systems they aim to represent, leading to more insightful analyses and ultimately, more accurate predictions.

The continual refinement of polynomial regression and linear basis function models is driving a significant leap forward in the field of regression analysis. These advancements aren’t simply about accommodating curves; they fundamentally enhance a model’s ability to represent intricate datasets and generalize beyond the training data. By moving beyond the limitations of linear relationships, researchers are creating tools capable of accurately predicting outcomes in complex systems – from financial markets and climate modeling to medical diagnostics and image recognition. This increased versatility stems from the models’ heightened capacity to capture nuanced patterns and interactions within data, ultimately leading to more reliable and insightful predictions and a broader range of applicable scenarios.

Increasing model complexity causes a transition from underfitting to overfitting.

The document champions a minimalist approach to modeling, starting with linear regression and progressively building complexity only as needed. This aligns perfectly with Kernighan’s sentiment: “Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you are, by definition, not smart enough to debug it.” The tutorial’s emphasis on regularization, designed to combat overfitting, embodies this principle. It acknowledges that a more complex model isn’t inherently better; rather, a simpler model, effectively tuned to generalize, demonstrates a deeper understanding of the underlying data and avoids unnecessary computational burden. The progression from linear models to neural networks isn’t about escalating complexity for its own sake, but about judiciously adding layers of abstraction only when simpler methods prove insufficient.

What Remains?

The proliferation of models does not equate to understanding. This document traces a lineage-from the readily interpretable line to the opaque depths of layered networks. The core challenge persists: minimizing error is trivial; minimizing meaningless error is not. Current efforts largely address the ‘how’ of prediction, postponing the ‘why’.

Future work will not be defined by novel architectures, but by rigorous constraint. The field chases complexity, yet often the most valuable insights arise from distilling phenomena to their essential components. A return to fundamental questions-what constitutes a good fit, and what information is truly extracted-is necessary.

The temptation to treat models as black boxes must be resisted. Interpretability is not merely a desirable feature; it is a prerequisite for genuine progress. The pursuit of accuracy, divorced from understanding, yields only increasingly elaborate, and ultimately fragile, approximations.

Original article: https://arxiv.org/pdf/2512.04747.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/