Gradient

See also: Machine learning terms

Gradient in Machine Learning

Gradient is a fundamental concept in machine learning, particularly in the optimization process of training algorithms. It is a multi-dimensional generalization of the derivative, which quantifies the rate of change of a function with respect to its variables. In machine learning, the gradient is typically used to minimize a loss function, which measures the discrepancy between the predictions made by the model and the actual target values.

Definition and Mathematical Representation

Formally, the gradient of a scalar-valued function, f(x), with respect to a vector variable, x ∈ ℝⁿ, is denoted by the symbol ∇f(x) and defined as a vector whose components are the partial derivatives of f with respect to each component of x. Mathematically, the gradient can be expressed as:

∇f(x) = [ ∂f/∂x₁, ∂f/∂x₂, ..., ∂f/∂xₙ ]ᵀ

where n represents the number of dimensions or variables in the vector x, and the superscript "ᵀ" denotes the transpose operation.

Application in Machine Learning

In machine learning, the gradient is primarily used in the context of gradient descent, an iterative optimization algorithm that seeks to minimize a given function. The algorithm updates the model's parameters iteratively, following the direction of the steepest descent, which is the negative of the gradient of the loss function with respect to the model's parameters:

θ = θ - η ∇L(θ)

where θ represents the model's parameters, L(θ) is the loss function, and η is the learning rate, a scalar that controls the step size of the updates.

The gradient is also central to other optimization techniques used in machine learning, such as stochastic gradient descent (SGD), momentum, and adaptive moment estimation (Adam).

Explain Like I'm 5 (ELI5)

Imagine you're on top of a hill and you want to reach the bottom as quickly as possible. Gradient in machine learning is like an arrow that points you in the direction where the hill is steepest, so you can slide down faster. By following the arrows (or gradients), you can eventually reach the bottom of the hill.

In machine learning, the hill is like a problem we want to solve, and the bottom is the best solution we can find. The arrows (gradients) help us figure out how to change our current solution to make it better, so we can find the best solution more quickly.