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Illustration of gradient descent on a series of level sets. Gradient descent is based on the observation that if the multi-variable function is defined and differentiable in a neighborhood of a point , then () decreases fastest if one goes from in the direction of the negative gradient of at , ().
The following figure illustrates an example of 2-dimensional Rosenbrock function optimization by adaptive coordinate descent from starting point = (,). The solution with the function value 10 − 10 {\displaystyle 10^{-10}} can be found after 325 function evaluations.
In short, because the gradient descent steps are too large, the variance in the stochastic gradient starts to dominate, and starts doing a random walk in the vicinity of . For decreasing learning rate schedule with η k = O ( 1 / k ) {\textstyle \eta _{k}=O(1/k)} , we have E [ f ( x k ) − f ∗ ] = O ( 1 / k ) {\displaystyle \mathbb {E} \left ...
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.
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 ]
Gradient descent (alternatively, "steepest descent" or "steepest ascent"): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems. Subgradient methods: An iterative method for large locally Lipschitz functions using generalized gradients. Following Boris T ...
The reparameterization trick (aka "reparameterization gradient estimator") is a technique used in statistical machine learning, particularly in variational inference, variational autoencoders, and stochastic optimization.
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.