Enter the current point in the sequence, step size, and gradient into the calculator to determine the next point in the sequence using steepest descent.
Steepest Descent Formula
The following formula is used to calculate the steepest descent in optimization problems.
X(k+1) = X(k) - α * ∇f(X(k))
Variables:
- X(k+1) is the next point in the sequence
- X(k) is the current point in the sequence
- α is the step size (also known as learning rate)
- ∇f(X(k)) is the gradient of the function at the current point
To calculate the next point in the steepest descent, subtract the product of the step size and the gradient of the function at the current point from the current point. The step size determines how big a step to take in the direction of the steepest descent. The gradient of the function at the current point indicates the direction of the steepest ascent, and subtracting it moves us in the direction of the steepest descent.
What is a 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.
How to Calculate Steepest Descent?
The following steps outline how to perform Steepest Descent using the given formula:
- First, determine the current point in the sequence, X(k).
- Next, calculate the gradient of the function at the current point, ∇f(X(k)).
- Next, determine the step size or learning rate, α.
- Using the formula X(k+1) = X(k) – α * ∇f(X(k)), calculate the next point in the sequence, X(k+1).
- Repeat steps 2-4 until the desired convergence or stopping criteria is met.
Example Problem:
Use the following variables as an example problem to test your knowledge:
X(k) = 3
α = 0.1
∇f(X(k)) = 2
