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The properties of gradient descent depend on the properties of the objective function and the variant of gradient descent used (for example, if a line search step is used). The assumptions made affect the convergence rate, and other properties, that can be proven for gradient descent. [33]
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.
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 ...
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 ...
As observed above, is the negative gradient of at , so the gradient descent method would require to move in the direction r k. Here, however, we insist that the directions must be conjugate to each other. A practical way to enforce this is by requiring that the next search direction be built out of the current residual and all previous search ...
Another way is the so-called adaptive standard GD or SGD, some representatives are Adam, Adadelta, RMSProp and so on, see the article on Stochastic gradient descent. In adaptive standard GD or SGD, learning rates are allowed to vary at each iterate step n, but in a different manner from Backtracking line search for gradient descent.
In machine learning, early stopping is a form of regularization used to avoid overfitting when training a model with an iterative method, such as gradient descent. Such methods update the model to make it better fit the training data with each iteration. Up to a point, this improves the model's performance on data outside of the training set (e ...
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 ...