When OLS Isn’t Enough: Exploring the Power of Gradient Descent in Linear Regression 📉✨

Linear Regression has always been a go-to for data scientists, with its elegant simplicity and straightforward interpretation. But what makes this technique truly work? It’s all about finding the right slope (M) and intercept (C). Typically, we turn to Ordinary Least Squares (OLS), a method that’s been guiding statisticians for ages with its direct, formulaic way of pinpointing these values.

So, what is OLS? OLS is a predictive model used to find the linear relationship between a dependent variable and one or more independent variables. It does this by minimizing the sum of squared differences between observed values and predicted values, effectively fitting a line through the data points.

Welcome to part 3 of Data Diaries by Divya, where we’re diving into the math and method behind finding the best fit.

Why OLS Isn’t Always the Best Choice

As popular as OLS is, it has its limitations. Here are some key challenges:

Assumptions Violations: OLS relies on assumptions like linearity, independence, homoscedasticity, and normality of errors. When these assumptions don’t hold, OLS can produce biased or misleading estimates.

Multicollinearity: Highly correlated predictors can inflate standard errors, making it harder to interpret results accurately.

Sensitivity to Outliers: Outliers can skew results in a big way, and OLS is particularly vulnerable to this.

Non-linearity: OLS works well with linear relationships, but for non-linear data, we need transformations or alternative methods.

Heteroscedasticity: When the error variance isn’t constant, OLS can provide inefficient or unstable estimates.

High Dimensionality: OLS is prone to overfitting when there are too many predictors relative to observations.

Non-Normal Errors: When errors aren’t normally distributed, confidence intervals and significance tests can be misleading.

Enter Gradient Descent: A Solution to OLS Limitations

When OLS starts to struggle, Gradient Descent steps in. Rather than making rigid assumptions, Gradient Descent takes an iterative approach, “stepping” its way toward the best values for M and C, even if the data doesn’t fit all of OLS’s requirements. It’s particularly useful in high-dimensional or non-linear settings where OLS falls short.

With each iteration, Gradient Descent refines the values of M and C by minimizing the error between predicted and actual values. This makes it a powerful alternative for handling complex datasets where OLS might stumble.

Wrapping Up

Linear Regression may look simple, but the techniques behind it reveal a lot about data’s complexity. OLS remains a trusted method for many, but knowing when to pivot to Gradient Descent is part of the data science journey. After all, our job as data scientists isn’t just to apply methods but to understand when to adapt them to our data.

Data science is all about choices, and picking the right technique is one of them!