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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 ]
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 ...
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.
(In Nocedal & Wright (2000) one can find a description of an algorithm with 1), 3) and 4) above, which was not tested in deep neural networks before the cited paper.) One can save time further by a hybrid mixture between two-way backtracking and the basic standard gradient descent algorithm.
Numerous methods exist to compute descent directions, all with differing merits, such as gradient descent or the conjugate gradient method. More generally, if P {\displaystyle P} is a positive definite matrix, then p k = − P ∇ f ( x k ) {\displaystyle p_{k}=-P\nabla f(x_{k})} is a descent direction at x k {\displaystyle x_{k}} . [ 1 ]
This includes, for example, early stopping, using a robust loss function, and discarding outliers. Implicit regularization is essentially ubiquitous in modern machine learning approaches, including stochastic gradient descent for training deep neural networks, and ensemble methods (such as random forests and gradient boosted trees).
The standard method for training RNN by gradient descent is the "backpropagation through time" (BPTT) algorithm, which is a special case of the general algorithm of backpropagation. A more computationally expensive online variant is called "Real-Time Recurrent Learning" or RTRL, [ 78 ] [ 79 ] which is an instance of automatic differentiation in ...
The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and reasonable starting parameters, the LMA tends to be slower than the GNA.