Steepest Descent Calculator

Formula

To calculate the next point in the steepest descent:

\[ X(k+1) = X(k) - \alpha \cdot \nabla f(X(k)) \]

Where:

\( X(k+1) \) is the next point in the sequence
\( X(k) \) is the current point in the sequence
\( \alpha \) is the step size (also known as learning rate)
\( \nabla f(X(k)) \) is the gradient of the function at the current point

What is Steepest Descent?

Steepest Descent is an iterative optimization algorithm used to find the local minimum of a function. It works by taking steps proportional to the negative of the gradient (or approximate gradient) of the function at the current point. The direction of the steepest descent is the direction of the negative gradient. This method is commonly used in machine learning and data analysis for optimizing cost functions.

Example Calculation

Let's assume the following values:

Current Point \( X(k) \) = 4
Step Size \( \alpha \) = 0.1
Gradient at Current Point \( \nabla f(X(k)) \) = 2

Step 1: Multiply the step size by the gradient at the current point:

\[ \alpha \cdot \nabla f(X(k)) = 0.1 \cdot 2 = 0.2 \]

Step 2: Subtract the result from the current point:

\[ X(k+1) = 4 - 0.2 = 3.8 \]