Introduction to Gradient Descent
Gradient descent is an optimization algorithm used to minimize a function by iteratively moving towards the steepest descent, as defined by the negative of the gradient. It is widely used in machine learning and deep learning to optimize model parameters.
Mathematical Formulation
The update rule for gradient descent is:
$$ \theta^{(t+1)} = \theta^{(t)} - \eta \nabla_\theta J(\theta) $$
where:
- \( \theta \) represents the parameters to be optimized.
- \( \eta \) is the learning rate, a hyperparameter that controls the step size of each update.
- \( J(\theta) \) is the cost function that measures the error of the model.
Simple Example
Consider minimizing \( f(x) = x^2 \). The gradient is \( \nabla f(x) = 2x \). The update rule becomes:
$$ x^{(t+1)} = x^{(t)} - \eta \cdot 2x^{(t)} $$
This process iteratively reduces \( x \) towards zero, the minimum of \( f(x) \).
Intermediate Example
Let's minimize \( f(x, y) = x^2 + y^2 \). The gradients are \( \nabla f(x) = 2x \) and \( \nabla f(y) = 2y \).
The update rules are:
$$ x^{(t+1)} = x^{(t)} - \eta \cdot 2x^{(t)} $$
$$ y^{(t+1)} = y^{(t)} - \eta \cdot 2y^{(t)} $$
This process iteratively reduces both \( x \) and \( y \) towards zero, the minimum of \( f(x, y) \).
Advanced Example
Consider a more complex function \( f(x, y) = 3x^2 + 4xy + y^2 \). The gradients are:
$$ \nabla f(x) = 6x + 4y $$
$$ \nabla f(y) = 4x + 2y $$
The update rules are:
$$ x^{(t+1)} = x^{(t)} - \eta (6x^{(t)} + 4y^{(t)}) $$
$$ y^{(t+1)} = y^{(t)} - \eta (4x^{(t)} + 2y^{(t)}) $$
This process iteratively adjusts both \( x \) and \( y \) to find the minimum of the function.
Visualizing Gradient Descent
Imagine standing on a hill and trying to find the lowest point. Gradient descent is like taking steps downhill, always moving in the direction of the steepest descent. The learning rate determines the size of each step.
Applications in Real Life
Gradient descent is used in various fields, including machine learning, economics, and physics. In machine learning, it is used to train models by minimizing the error between predicted and actual values. In economics, it can be used to optimize cost functions for better decision-making.