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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.
This technique is used in stochastic gradient descent and as an extension to the backpropagation algorithms used to train artificial neural networks. [29] [30] In the direction of updating, stochastic gradient descent adds a stochastic property. The weights can be used to calculate the derivatives.
In the stochastic setting, under the same assumption that the gradient is Lipschitz continuous and one uses a more restrictive version (requiring in addition that the sum of learning rates is infinite and the sum of squares of learning rates is finite) of diminishing learning rate scheme (see section "Stochastic gradient descent") and moreover ...
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.
The algorithm starts with an initial estimate of the optimal value, , and proceeds iteratively to refine that estimate with a sequence of better estimates ,, ….The derivatives of the function := are used as a key driver of the algorithm to identify the direction of steepest descent, and also to form an estimate of the Hessian matrix (second derivative) of ().
SGLD can be applied to the optimization of non-convex objective functions, shown here to be a sum of Gaussians. Stochastic gradient Langevin dynamics (SGLD) is an optimization and sampling technique composed of characteristics from Stochastic gradient descent, a Robbins–Monro optimization algorithm, and Langevin dynamics, a mathematical extension of molecular dynamics models.
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).
This is based on the gradient descent algorithm. The algorithm starts by assuming small weights (zero in most cases) and, at each step, by finding the gradient of the mean square error, the weights are updated.
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