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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.
In the stochastic setting (such as in the mini-batch setting in deep learning), standard GD is called stochastic gradient descent, or SGD. Even if the cost function has globally continuous gradient, good estimate of the Lipschitz constant for the cost functions in deep learning may not be feasible or desirable, given the very high dimensions of ...
The number of gradient descent iterations is commonly proportional to the spectral condition number of the system matrix (the ratio of the maximum to minimum eigenvalues of ), while the convergence of conjugate gradient method is typically determined by a square root of the condition number, i.e., is much faster.
Mini-batch techniques are used with repeated passing over the training data to obtain optimized out-of-core versions of machine learning algorithms, for example, stochastic gradient descent. When combined with backpropagation, this is currently the de facto training method for training artificial neural networks.
It allows for the efficient computation of gradients through random variables, enabling the optimization of parametric probability models using stochastic gradient descent, and the variance reduction of estimators. It was developed in the 1980s in operations research, under the name of "pathwise gradients", or "stochastic gradients".
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.
Federated stochastic gradient descent [19] is the direct transposition of this algorithm to the federated setting, but by using a random fraction of the nodes and using all the data on this node. The gradients are averaged by the server proportionally to the number of training samples on each node, and used to make a gradient descent step.
One can also apply a widespread stochastic gradient descent method with iterative projection to solve this problem. [6] The idea of this method is to update the dictionary using the first order stochastic gradient and project it on the constraint set . The step that occurs at i-th iteration is described by this expression: