Understanding the Mathematics Behind Machine Learning: Gradient Descent
“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.” — William Paul Thurston
Every ML algorithm, from linear regression to deep neural networks, is built upon core mathematical principles. These include statistics, probability, calculus, and linear algebra, all of which help machines learn patterns, make predictions, and improve over time.
When it comes to mastering machine learning, understanding the fundamental mathematical principles that underpin algorithms like Gradient Descent is essential. Most online courses focus on the application of Gradient Descent in ML without delving into the core mathematical concepts behind it.
Before we explore the details of Optimization using Gradient Descent, let’s first understand the basics of model training. A dataset used for training a machine learning algorithm consists of input data (features) and corresponding output data (targets). During training, the model learns to make predictions by comparing its outputs to the ground truth provided in the dataset.
To minimize errors and optimize model performance, it is crucial to find the optimal parameters such as weights and biases. This process, known as optimization, is facilitated by mathematical tools like Gradient Descent, which plays a key role in training supervised models effectively.
Optimization and Gradient Descent
Optimization in mathematical terms involves minimizing or maximizing a function by adjusting its variables. For Gradient Descent to work efficiently, the objective function must be differentiable and convex. Differentiability ensures smoothness, while convexity guarantees the existence of a global minimum.
Through Gradient Descent, we calculate the gradients of the loss function and iteratively adjust the model’s parameters to converge to the global minimum. This iterative process allows the model to improve its predictions by minimizing the loss.
By understanding the mathematical foundations of Gradient Descent, data scientists and engineers can design more effective models and interpret their results with greater precision.
Implementation of Gradient Descent in Machine Learning
In machine learning, the objective function is defined as y_pred = f(x;θ), where ‘y_pred’ is the predicted value, ‘x’ represents the inputs, and ‘θ’ denotes the parameters (weights and biases). The goal is to find optimal weights and biases that correspond to minimum loss.
By calculating the gradients of the loss function with respect to the parameters, we can update the weights in each iteration to minimize the loss. This step-by-step process of parameter optimization is the essence of the Gradient Descent algorithm.
The learning rate, a hyperparameter in Gradient Descent, controls the size of parameter updates and plays a crucial role in determining the convergence speed and stability of the algorithm.
Overall, understanding the mathematics behind Gradient Descent is essential for anyone working in machine learning, as it forms the foundation for optimizing models and improving their performance.