gradientbasert

Gradientbasert refers to methods that rely on gradient information of a function to guide optimization, learning, or analysis. It is a cross-disciplinary concept used in mathematics, computer science, engineering, and statistics.

In optimization, gradient-based methods compute the gradient vector of the objective function with respect to decision variables and move parameters in the direction of the negative gradient to reduce the objective. The basic scheme is x_{k+1} = x_k - eta grad f(x_k). They assume differentiability and access to gradient calculations. Variants include stochastic gradient descent and mini-batch SGD, often combined with momentum, adaptive learning rates (Adam, RMSprop, Adagrad).

In machine learning, gradient-based optimization is used to train models by minimizing loss functions; backpropagation computes gradients efficiently through the computational graph. Automatic differentiation tools in frameworks such as TensorFlow, PyTorch, and JAX support gradient computations on large models.

Other uses include gradient-based feature extraction and image processing, where gradient magnitude or orientation informs edge detection and tracking.

Advantages include scalability to high-dimensional problems and strong performance on differentiable objectives; limitations include susceptibility to local minima and saddle points, sensitivity to hyperparameters, and the need for differentiability and gradient evaluation costs.

History and related terms: the steepest descent method, introduced by Cauchy, is an early gradient-based algorithm; modern variants include conjugate gradient and various momentum-based methods; gradient-free methods such as Nelder-Mead provide alternatives. Notable tools include TensorFlow, PyTorch, and JAX.