Optimization using Gradient Descent in one variable

Gradient Descent is an iterative method extensively utilized in finding the minimum or maximum of functions, particularly beneficial in multi-variable scenarios. This approach is foundational in optimization problems where exact solutions are challenging to derive analytically, especially due to complexities in higher dimensions.

Concept of Gradient Descent

• Initially demonstrated in a single-variable context to ease into the more complex multi-variable gradient descent.
• Employs an iterative process to approximate the minimum of a function by systematically updating the point of interest based on the function's derivative.

Mathematical Formulation

Given a function $f(x) = e^x - \log(x)$, finding its minimum involves:

1. Derivative Calculation: The first step is to compute the derivative $f'(x) = e^x - \frac{1}{x}$.
2. Iterative Update: Starting from an initial point $x_0$, the next point $x_{1}$ is determined by $x_{1} = x_{0} - \alpha \cdot f'(x_{0})$, where $\alpha$ is the learning rate.

This method leverages the derivative to guide the direction of steps taken towards the minimum. The sign of the derivative indicates whether to move left or right (in the case of a single variable).

Challenges and Solutions

Analytical Difficulty

• Directly solving $e^x = \frac{1}{x}$ for $x$ is analytically challenging, exemplifying situations where gradient descent offers a practical solution.

Learning Rate ($\alpha$)

• The learning rate $\alpha$ is critical in ensuring the iterative steps are appropriately sized to prevent overshooting or excessively slow convergence.
• Adaptive Learning Rates: Research into adaptive learning rates seeks to dynamically adjust $\alpha$ based on the optimization progress, though a universally optimal strategy is yet to be established.

Local Minima

• Gradient descent may converge to local minima, potentially missing the global minimum.
• Multiple Initial Points: Employing multiple starting points and running gradient descent iterations from each can enhance the likelihood of approaching the global minimum.

Practical Implementation

1. Initialization: Choose a starting point and learning rate.
2. Update Rule: Apply the update $x_{k} = x_{k-1} - \alpha \cdot f'(x_{k-1})$ iteratively.
3. Convergence Criterion: Iterations continue until changes in the value of $x$ become negligible, indicating proximity to the minimum.

Example

• For $f(x) = e^x - \log(x)$, starting from an initial guess, iterations proceed by computing the derivative at the current point and updating the point according to the learning rate and the computed gradient.
• This process does not require solving for when the derivative equals zero but rather iteratively adjusts based on the gradient's direction and magnitude.

Conclusion

Gradient Descent is a powerful tool for optimization, particularly in contexts where analytical solutions are impractical. Its effectiveness hinges on the careful selection of the learning rate and initial points, illustrating both its utility and complexity in machine learning and optimization tasks.