Gradient descent is a fundamental optimization algorithm used to find the minimum of a function. The core idea is simple: start at any point, calculate the gradient which tells us the direction of steepest increase, then move in the opposite direction toward the steepest decrease. Let's see this in action with a simple quadratic function.
The mathematical formula for gradient descent is theta new equals theta old minus alpha times the gradient of f at theta. Here, theta represents our parameters, alpha is the learning rate that controls step size, and the gradient tells us the direction. Let's see how this formula works step by step, starting from an initial point and taking successive steps toward the minimum.
The learning rate alpha is crucial for gradient descent performance. If alpha is too large, like two point zero, we overshoot the minimum and may even diverge. If alpha is too small, like zero point one, we make very slow progress requiring many iterations. The optimal learning rate, like zero point five, provides steady progress with fast convergence. Let's see how different learning rates affect the path to the minimum.
A major challenge in gradient descent is dealing with local versus global minima. In complex functions with multiple valleys, gradient descent can get trapped in local minima - points that are lowest in their neighborhood but not the global optimum. Starting from different initial points can lead to different local minima. This is why we often use multiple random starting points or advanced algorithms to find the global minimum.
To summarize gradient descent: it finds function minima by iteratively moving in the direction of steepest descent. The update rule combines gradient direction with learning rate to control step size. Proper learning rate selection balances speed and stability. While local minima present challenges, the core principle remains fundamental to machine learning and optimization across many domains.