Gradientedbaserade

Gradientedbaserade, in English gradient-based, refers to methods that use the gradient information of a scalar objective function to guide updates during optimization or learning. The gradient points in the direction of steepest increase, so minimization typically proceeds by stepping in the opposite direction. Gradient-based methods rely on differentiability and access to gradient computations, which can be exact analytically, numerically, or via automatic differentiation.

Common gradient-based optimization methods include plain gradient descent, stochastic gradient descent (SGD) and its variants with momentum or adaptive learning rates; Nesterov accelerated gradient; and quasi-Newton methods such as BFGS and L-BFGS that use gradient information together with curvature approximations. These methods differ in how they sample data, estimate the gradient, and adjust step sizes.

Automatic differentiation enables efficient and accurate gradient computation in complex models, notably deep neural networks where backpropagation computes gradients through many layers. Convergence properties depend on problem structure: in convex problems, gradient-based methods can reach the global optimum; in non-convex settings they typically reach a local minimum or saddle point, with performance influenced by learning rate, initialization, and regularization.

Applications span machine learning, scientific computing, engineering, and control. Practical issues include ill-conditioning, noisy gradients, and saddle points. Techniques such as gradient clipping, normalization, learning-rate schedules, and adaptive optimizers help improve stability and speed. Gradient-based methods contrast with gradient-free approaches that do not require gradient information.