Comparison of Two Linear Regression Methods: Least Squares and Gradient Descent

Linear regression is a machine learning technique that is widely used to solve real-world problems. It is simple yet powerful and can be applied to a variety of datasets. There are different methods of implementing linear regression, but two popular ones are least squares and gradient descent. In this blog post, we will compare these two methods and discuss their pros and cons.

Least Squares

Least squares is a classic method used to fit a line to a set of data points. It is also known as the ordinary least squares (OLS) method. In this method, we try to minimize the sum of the squared differences between the predicted values and the actual values. Mathematically, the equation for a simple linear regression using OLS is as follows:

Y = β0 + β1*X + ε

where Y is the dependent variable, X is the independent variable, β0 and β1 are the coefficients, and ε is the error term. The coefficients can be calculated using the following formulas:

β1 = Σ((xi - x_mean)*(yi - y_mean))/Σ(xi - x_mean)^2
β0 = y_mean - β1*x_mean

where xi and yi are the values of the independent and dependent variables, respectively, x_mean and y_mean are their means, and Σ denotes summation.

Gradient Descent

Gradient descent is an iterative method used to minimize the cost function in machine learning algorithms. In linear regression, the cost function is the sum of the squared differences between the predicted values and the actual values, just like in least squares. The difference is that instead of directly calculating the coefficients, we use an iterative algorithm to update them until we reach the minimum.

In gradient descent, we start with random initial values for the coefficients, and then we update them using the following equations:

β1 = β1 - α*Σ((φ(xi) - yi)*xi)
β0 = β0 - α*Σ(φ(xi) - yi)

where φ(xi) is a feature function that transforms the input data into a higher dimensional space, α is the learning rate, and Σ denotes summation.

Comparison

Now, let's compare the two methods based on their pros and cons.

Pros of Least Squares

• It has a closed-form solution, which means that the coefficients can be calculated directly without the need for an iterative algorithm.
• It is computationally efficient for small datasets or problems with a small number of variables.

Cons of Least Squares

• It can be sensitive to outliers in the data, which can lead to overfitting.
• It does not work well with large datasets or problems with a large number of variables, as the calculations can become computationally expensive.

Pros of Gradient Descent

• It is more robust to outliers, as it iteratively updates the coefficients based on a subset of the data.
• It can handle large datasets or problems with a large number of variables, as the calculations can be distributed across multiple processors.

Cons of Gradient Descent

• It requires tuning of the learning rate parameter, which can affect the convergence of the algorithm.
• It can converge to a local minimum instead of the global minimum, depending on the initial values and the learning rate.

Conclusion

In conclusion, both least squares and gradient descent are effective methods for implementing linear regression. The choice of method depends on the specific problem at hand, the size of the dataset, and the desired level of accuracy. Least squares is a good option for small datasets and problems with a simple structure, while gradient descent is better suited for large datasets and problems with more complex patterns.

References

• Hastie, T., Tibshirani, R., & Friedman, J. (2017). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition. Springer.
• James, G., Witten, D., Hastie, T., & Tibshirani, R. (2017). An Introduction to Statistical Learning: with Applications in R. Springer.