Gradient descent is an optimization algorithm used to find the minimum of a function. In machine learning, it's commonly used to minimize the cost or loss function of a model by iteratively adjusting the model's parameters, such as weights and biases. The algorithm starts at an arbitrary point on the function's surface and calculates the gradient, which indicates the direction of steepest increase. It then updates the parameters by moving in the opposite direction of the gradient - the direction of steepest decrease.
Let's break down the gradient descent algorithm step by step. First, we calculate the gradient or slope of the cost function at our current position. The gradient is a vector that points in the direction of the steepest increase. Second, we move in the opposite direction of the gradient, which is the direction of steepest decrease. The size of our step is determined by the learning rate alpha. Finally, we repeat this process until we reach a minimum where the gradient is close to zero. The update rule is shown by this formula, where w represents our parameters, alpha is the learning rate, and the partial derivative represents the gradient.
The learning rate is a crucial hyperparameter in gradient descent that controls the step size. If the learning rate is too small, the algorithm will converge very slowly, requiring many iterations to reach the minimum. If the learning rate is too large, the algorithm may overshoot the minimum or even diverge, bouncing back and forth or moving away from the solution. An optimal learning rate allows for efficient convergence, reaching the minimum in a reasonable number of steps. Many advanced optimization algorithms use adaptive learning rates that adjust automatically during training to improve convergence speed and stability.
There are several variants of gradient descent that differ in how much data they use per iteration. Batch gradient descent uses all training examples to compute the gradient before performing an update. This provides stable and accurate updates but can be computationally expensive for large datasets. Stochastic gradient descent, or SGD, uses just one randomly selected training example to compute the gradient for each update. This makes each iteration much faster, but the updates are noisy and can cause the path to minimum to be less direct. Mini-batch gradient descent strikes a balance by using small random batches of examples for each update, combining the stability of batch methods with the efficiency of stochastic approaches. This is the most commonly used variant in practice.
Beyond basic gradient descent, several advanced optimization algorithms have been developed to improve training efficiency. Momentum adds a velocity component to the updates, helping to smooth out oscillations and accelerate convergence, especially in ravines where the gradient changes direction frequently. RMSprop adapts the learning rate for each parameter based on the history of gradients, allowing larger updates for infrequent parameters and smaller updates for frequent ones. Adam, which stands for Adaptive Moment Estimation, combines the benefits of both momentum and adaptive learning rates, making it one of the most popular optimizers today. These advanced algorithms help navigate complex loss landscapes with multiple local minima, saddle points, and plateaus, significantly speeding up convergence compared to standard gradient descent.