Gradient Descent is an optimization algorithm used to find the minimum of a function by iteratively moving in the direction of steepest decrease. In machine learning, it's commonly used to minimize the cost or loss function and find optimal model parameters. This visualization shows a simple cost function with two parameters. The algorithm starts at an initial point and follows the negative gradient, which points in the direction of steepest descent, eventually converging to the minimum of the function.
Let's break down the gradient descent algorithm step by step. First, we initialize the parameters with some values. Then, we calculate the gradient of the cost function with respect to each parameter. The gradient is a vector that points in the direction of the steepest increase of the function. Next, we update the parameters by moving in the opposite direction of the gradient. The step size is determined by the learning rate alpha. Finally, we repeat this process until convergence. The update rule is: theta new equals theta old minus alpha times the gradient of J of theta. In this visualization, the red dot represents our current parameter value, the yellow arrow shows the gradient direction, and the red arrow shows the update direction. As we follow the negative gradient, we move closer to the minimum of the cost function.
Let's explore the common variants of gradient descent. Batch gradient descent uses the entire dataset to compute the gradient for each parameter update. This provides a stable path to the minimum but can be computationally expensive for large datasets. Stochastic gradient descent, or SGD, uses just one randomly selected sample to compute the gradient for each update. This makes it much faster but results in a noisier convergence path with more fluctuations. Mini-batch gradient descent strikes a balance between these approaches by using small random batches of data for each update. This provides a good compromise between computation speed and convergence stability. In the visualization, you can see how batch gradient descent follows a direct path to the minimum, SGD takes a noisy zigzag path, and mini-batch follows a moderately noisy path.
Now let's explore some advanced gradient descent techniques that improve upon the basic algorithm. Momentum is a technique that adds a fraction of the previous update to the current one. This helps accelerate convergence and navigate through ravines and local minima more effectively. The update rule includes a velocity term that accumulates gradients over time. AdaGrad adapts the learning rate for each parameter based on its historical gradients. Parameters that are updated infrequently receive larger updates, while frequently updated parameters get smaller updates. This is particularly useful for sparse data. Adam combines the benefits of momentum and adaptive learning rates. It maintains both a decaying average of past gradients and past squared gradients to adapt the learning rate for each parameter. Adam is often the default choice for deep learning applications due to its robust performance. In the visualization, you can see how these advanced techniques converge faster to the local minimum compared to standard gradient descent.
To summarize what we've learned about gradient descent: It's an iterative optimization algorithm that finds the minimum of a function by following the direction of the negative gradient. The basic variants include Batch Gradient Descent, which uses the entire dataset; Stochastic Gradient Descent, which uses a single random sample; and Mini-batch Gradient Descent, which uses small random batches. Advanced techniques like Momentum, AdaGrad, and Adam improve convergence speed and stability. Gradient descent is widely used in machine learning to minimize cost functions and find optimal model parameters. Understanding this algorithm is fundamental to working with many machine learning models, especially neural networks and deep learning architectures.