Difference between Gradient Descent and Normal Equation


When it comes to understanding regression issues in machine learning, two commonly utilized procedures are gradient descent and the normal equation. Whereas both strategies point to discover the ideal parameters for a given demonstrate, they take unmistakable approaches to realize this objective. Gradient descent is an iterative optimization calculation that steadily alters the parameters by minimizing the cost function, whereas the normal equation gives a closed−form solution straightforwardly. Understanding the contrasts between these two approaches is vital in selecting the foremost suitable method for a specific issue. In this article, we'll dig into the incongruities between gradient descent and the normal equation, highlighting their qualities and shortcomings.

Gradient Descent

It is an iterative optimization algorithm broadly utilized in machine learning and mathematical optimization. Its primary objective is to minimize a cost function by iteratively altering the parameters of a model. The calculation is especially successful in finding ideal parameter values for models such as linear regression, logistic regression, and neural networks.

The concept behind gradient descent is to require steps within the direction of steepest descent within the cost function scene. The "gradient" alludes to the subordinate of the cost function with respect to each parameter. By calculating the slope, the calculation determines the direction of the steepest increase within the fetched function and after that alters the parameters within the inverse direction to minimize the cost.

The method starts with the initialization of the parameters with random or predefined values. In each iteration, the calculation computes the cost function, which speaks to the error between the anticipated and actual values. At that point calculates the angle of the cost function about each parameter.

Angle descent could be a powerful optimization strategy that permits models to memorize from information and discover the ideal parameters for achieving exact predictions.

Normal Equation

The normal equation is a closed−form solution utilized in linear regression to discover the ideal parameters that minimize the entirety of squared errors. Not at all like gradient descent, which is an iterative optimization calculation, the normal condition gives a direct solution without the requirement for iterative upgrades.

The concept behind the normal equation is to require the subsidiary of the cost function about the parameters and set it to zero. By doing so, we are able discover the parameter values that fulfill this condition and minimize the cost function.

To get the optimal parameter values, the normal equation includes matrix operations such as matrix transposition and reversal. It calculates the parameters straightforwardly by duplicating the converse of the item of the transposed design matrix and the design matrix with the transposed design matrix and the target values.

The normal equation is especially valuable for straightforward linear regression issues with a single free variable, because it gives an exact solution. In any case, it can too be expanded to handle multiple independent variables.

Whereas the normal equation offers an efficient and exact solution, it can get to be computationally expensive for large datasets since it requires altering a matrix, which includes a time complexity of O(n^3). Also, it is particular to linear regression and cannot be specifically applied to other models.

Difference between Gradient Descent and Normal Equation

The differences are highlighted in the following table

Basis of Difference

Gradient descent

Normal equation





It may converge gradually

It converges straightforwardly


It is effective and productive for large datasets

It is inefficient for large datasets


It is slower for complex models

It is quicker for simple linear regression

Learning Rate

It must be carefully chosen

It is not applicable

Model Adaptability

It is applicable to different models

It is restricted to linear regression


It may get stuck in neighbourhood optima

It is steady for most cases


It is appropriate for large datasets.

It is constrained by matrix inversion for large datasets.


It underpins regularization methods.

It requires alteration for regularization

Include Scaling

It may require include scaling.

It is not influenced by highlight scaling


Gradient descent and the normal equation speak to diverse methodologies for optimizing parameters in machine learning. Gradient descent may be an adaptable iterative approach that can handle huge datasets and different models, but it may converge gradually and requires cautious tuning of the learning rate. On the other hand, the normal equation gives an exact solution for linear regression issues without the required for iterations, but it can be computationally costly for expansive datasets and is constrained to linear regression. The choice between gradient descent and the normal equation depends on the issue, dataset measure, computational assets, and the trade−off between precision and productivity.

Updated on: 28-Jul-2023


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