What Is Gradient Descent?

Gradient descent is an optimization algorithm in machine learning (ML) that aims to find the optimal parameter set for another machine learning model by iteratively minimizing the model's cost function. The cost function represents the discrepancy between the model's predicted data points and the corresponding real-world data points. The gradient descent algorithm attempts to find model parameters that minimize discrepancy and improve the model's performance.

The gradient descent algorithm is applicable in mathematics, statistics, data science, and machine learning (ML). It improves model accuracy over time by iteratively adjusting the model parameters to reduce the loss. It also plays a crucial role in training neural networks. It updates the network's weights by calculating the gradient of the loss function with respect to each weight. This process helps fine-tune the network's parameters to better capture the underlying patterns in the training data.

You can use gradient descent to solve optimization problems in different fields. We give some examples below.

Engineering

Gradient descent is applicable in various engineering and physics problems that involve minimizing energy, cost, or error. For instance, it can be used to find the optimal design of a structure that minimizes material cost while maximizing strength.

Finance

You can use gradient descent to optimize investment portfolios, minimize cost functions in production, or maximize utility functions in consumer choice models. It provides a methodical approach to finding the best resource allocation or the optimal price setting to achieve desired outcomes.

Image processing

Gradient descent is useful in image and signal processing for tasks like denoising, sharpening, and reconstructing images. It helps adjust image parameters to minimize the difference between the processed image and an ideal target image. This can improve image quality or extract important features.

Real-world data approximately fits statistical models, which are then used to predict future or unknown results. Gradient descent ensures that the parameters of the given statistical model result in the most accurate fit to the patterns in the real-world data.

The statistically fit model can be expressed as a mathematical equation or function. The gradient descent algorithm achieves the model's best fit by finding the model's optimal parameters and minimizing the model's cost function.

Cost function

A cost function calculates the average error across a statistically fit model. It is the difference between the real-world data points and the model-generated data points.

To find the cost function, you take the average of the differences between the values of each real-world data point and its predicted value as plotted along the statistical function line. For a simple example, when doing linear regression, the cost function is the average of the real data points' distances from the straight line.

There are various ways to calculate the cost function, depending on the type of statistical analysis at hand. For regression tasks, mean squared error is common. For classification tasks, you might use binary cross entropy. Hinge loss is a typical cost function method in support vector machine ML tasks. However, gradient descent only works when a model's cost function is differentiable and results in a convex shape.

Gradient

The gradient descent algorithm takes a cost function and calculates its gradient. When the model is a basic linear regression function, there are only two parameters: the slope and the intercept. If you plot the cost function on a graph, the gradient represents the slope or steepness of the line. Mathematically, the gradient is a derivative of the cost function at a given point on the line.

Algorithm steps

Gradient descent means bringing the gradient value as close to zero as possible. The gradient descent algorithm iteratively samples the gradient of the model's cost function for each of the model's parameters, trying to reach zero. There may be hundreds of parameters to consider in more complex machine learning models, such as neural network models.

Steps include:

1. Assign random values as the parameters of the model.
2. Given the parameters, calculate the predicted outputs of the model vs. the real outputs.
3. Calculate the gradient of the cost function for each parameter.
4. Make a scaled step in the opposite direction to attempt to minimize the gradient with the new parameters.
5. Repeat steps two to four until either the smallest gradient is found or the maximum iteration count is reached.

Learning rate

The learning rate in gradient descent is the size of the steps taken to reach the minimum. Data scientists typically start with a small value and evaluate and update it based on the behavior of the cost function. The learning rate is also called step size or alpha.

Gradient descent varies depending on the statistical model being used.

Batch gradient descent

Batch gradient descent is the standard version of the algorithm, where the error calculations are performed across the whole dataset. This type of gradient descent works best with datasets containing fewer real-world data points.

Stochastic gradient descent

When dealing with large datasets, computing gradient descent with all the data can take a really long time. Stochastic gradient descent (SGD) instead takes a small, randomized training example from within the dataset for each iteration of the algorithm. SGD is often more efficient and cost-effective for large datasets, although it can struggle to find the minimum as it fluctuates with its frequent updates.

Mini batch gradient descent

The mini batch gradient descent is a combination of batch gradient descent and stochastic gradient descent. A number of training examples are batched together, the steps run, and the parameters are updated. In practice, mini batch gradient descent is the best fit for most problems.

Gradient boosting tree

Gradient boosting trees are not a true type of gradient descent, although they work in a similar mechanism to the gradient descent algorithm. Gradient boosting trees build decision tree predictive models sequentially, calculating the differences between real-world and predicted data points to converge on a stronger final model.

Gradient descent is simple, easy to implement, and understand. It is highly scalable and capable of handling complex models with many more parameters and larger datasets.

However, it is sensitive to the choice of the learning rate. If the learning rate is too high, the algorithm might overshoot the minimum; if it's too low, performance may drop. It then becomes computationally expensive and time-consuming for large datasets.

Additionally, when the slope of the cost function is at or close to zero, the model stops learning. In some scenarios, the gradient descent gets stuck at an incorrect local minimum or saddle point that does not represent the full dataset.

Amazon SageMaker is a fully managed service for preparing data and building, training, and deploying machine learning (ML) models for any use case with fully managed infrastructure, tools, and workflows. SageMaker JumpStart provides pre-trained, open-source models for many problem types to help you get started with machine learning. You can incrementally train and tune these models before deployment.

The Amazon SageMaker linear learner algorithm provides a solution for both classification and regression problems. With the linear learner algorithm, you train with a distributed implementation of stochastic gradient descent (SGD). You can control the optimization process by choosing the optimization algorithm. For example, you can choose to use Adam, AdaGrad, stochastic gradient descent, or other optimization algorithms. XGBoost is another popular and efficient open-source implementation of the gradient-boosted trees algorithm.

Get started with the gradient descent algorithm on AWS by creating a free account today.