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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 ...
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 ...
A comparison of the convergence of gradient descent with optimal step size (in green) and conjugate vector (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2).
In optimization, a gradient method is an algorithm to solve problems of the form min x ∈ R n f ( x ) {\displaystyle \min _{x\in \mathbb {R} ^{n}}\;f(x)} with the search directions defined by the gradient of the function at the current point.
Choosing a proportionality constant and eliminating the minus sign to enable us to move the weight in the negative direction of the gradient to minimize error, we arrive at our target equation: = ′ ().
Gradient descent minimization is one of the simplest optimizations which can be used to minimize snake energy. [4] Each iteration takes one step in the negative gradient of the point with controlled step size to find local minima. This gradient-descent minimization can be implemented as
The minimization can be performed either by gradient descent, as proposed initially, or by other means, usually involving iterative methods. The number of iterations needs to be experimentally determined and convergent solutions are not always guaranteed. Many implementations prefer to use the first Principal Components as a starting configuration.
In optimization, a descent direction is a vector that points towards a local minimum of an objective function :.. Computing by an iterative method, such as line search defines a descent direction at the th iterate to be any such that , <, where , denotes the inner product.