Gradient descent is one of the main tools that is used in many machine learning and neural network algorithms. It acts as a guide to finding the minimum of a function. But what’s happening under the hood, and why do we need it?
The Core Idea: Going Downhill
Imagine you’re standing on top of a mountain. The weather is foggy, and you can barely see your surroundings. You want to reach the lowest point in the valley as quickly as possible without injuring yourself. However, all you can sense is the slope beneath your feet. To make progress, you’d take small steps downhill in the steepest direction. Similarly, gradient descent works by calculating the steepest direction for each step defined by the function’s gradient.
The gradient, a.k.a. a vector of partial derivatives, shows how the loss function changes with each parameter. By iterating over small steps downhill, we reduce the loss and move closer to the function’s minimum, optimizing our model’s parameters.
Why Not Just Solve for Zero?
A natural question comes up: if we want to minimize the loss, why not set each partial derivative to zero and solve? Wouldn’t that give us the minimum faster? Unfortunately, this approach isn’t typically feasible in the context of modern neural network models:
- High Complexity: In a deep neural network with many parameters, the loss function is often a very complex, high-dimensional function. Setting each partial derivative to zero would require solving a system of equations with hundreds, thousands, or even millions of variables, which is generally computationally infeasible. The complexity even grows significantly with the number of parameters, making a direct solution impractical.
- Multiple Local Minima: The loss functions for deep neural networks typically have multiple local minima. Equating derivatives to zero can lead to finding a point where the gradient is zero but it’s not a global minimum but a local minimum. On the other hand, gradient descent allows iterative exploration of the loss landscape. In practice, this approach often yields better results by finding a useful minimum, even if it’s not the absolute global minimum.
- Efficiency through Iteration: Gradient descent allows for finding approximate solutions iteratively rather than directly jumping to it. Iterative methods are advantageous because they provide a way to control convergence by adjusting the learning rate and can be run for a predetermined number of steps to manage computational resources. Additionally, methods like stochastic gradient descent (SGD), compute gradients using single random input rather than the entire dataset, which introduces some noise. This noise can help the algorithm escape local minima, potentially improving the generalization of the model.
Types of Gradient Descent and Why They Matter
Gradient descent has various forms, each suited for different tasks:
- Batch Gradient Descent: It is the original form of gradient descent, where it uses the entire training dataset in each update step. It ensures the gradient calculation is based on all available data points at once. The calculated gradient is an exact representation of the loss function’s slope, which leads to more stable and reliable updates. Additionally, it always converges to the same result, making it predictable. On the hand, this method is heavily computationally intensive and memory-demanding. It requires processing of the entire dataset for each update and also the entire dataset should fit in memory, which may not be feasible for very large datasets.
- Stochastic Gradient Descent (SGD): Here “stochastic” means random. Instead of computing loss for the entire training dataset, SGD chooses a single random example at each step. It then computes the loss and update parameters. SGD uses minimal resources since it processes a small amount of data at a time. Additionally, the randomness may help it escape local minima, but it introduces noise, so the path toward the minimum may zigzag. Also because of this noise, it might not converge smoothly and can oscillate around the minimum instead of settling.
- Mini-Batch Gradient Descent: Balances between the batch and stochastic approaches, splitting the dataset into small random subsets called mini-batches. For each mini-batch, the gradient of the loss function is calculated and the parameters are updated based on the average gradient computed over the mini-batch. The randomness helps avoid local minima while keeping computation more manageable than batch gradient descent. Noise still exists, but convergence is smoother than SGD.
Gradient Descent Works
Gradient descent is a robust tool, balancing thousands of interdependent parameters to find the minimum of complex, high-dimensional loss functions. By iteratively adjusting each parameter in concert, gradient descent leads a path toward optimization. While it’s not without limitations, gradient descent remains one of the most essential and elegant tools in neural networks.