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Gradient descent with momentum remembers the solution update at each iteration, and determines the next update as a linear combination of the gradient and the previous update. For unconstrained quadratic minimization, a theoretical convergence rate bound of the heavy ball method is asymptotically the same as that for the optimal conjugate ...
In optimization, a gradient method is an algorithm to solve problems of the form with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradient.
By contrast, gradient descent methods can move in any direction that the ridge or alley may ascend or descend. Hence, gradient descent or the conjugate gradient method is generally preferred over hill climbing when the target function is differentiable. Hill climbers, however, have the advantage of not requiring the target function to be ...
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the ...
While it is sometimes possible to substitute gradient descent for a local search algorithm, gradient descent is not in the same family: although it is an iterative method for local optimization, it relies on an objective function’s gradient rather than an explicit exploration of the solution space.
Stochastic gradient descent competes with the L-BFGS algorithm, [citation needed] which is also widely used. Stochastic gradient descent has been used since at least 1960 for training linear regression models, originally under the name ADALINE. [25] Another stochastic gradient descent algorithm is the least mean squares (LMS) adaptive filter.
The line-search method first finds a descent direction along which the objective function will be reduced, and then computes a step size that determines how far should move along that direction. The descent direction can be computed by various methods, such as gradient descent or quasi-Newton method. The step size can be determined either ...
The Barzilai-Borwein method [1] is an iterative gradient descent method for unconstrained optimization using either of two step sizes derived from the linear trend of the most recent two iterates. This method, and modifications, are globally convergent under mild conditions, [ 2 ] [ 3 ] and perform competitively with conjugate gradient methods ...