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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 , ().
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).
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.
A comparison of gradient descent (green) and Newton's method (red) for minimizing a function (with small step sizes). Newton's method uses curvature information (i.e. the second derivative) to take a more direct route.
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.
The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f ( r ) {\displaystyle f(\mathbf {r} )} may be defined by:
By defining the sequence + = and using the above identity, we can interpret the proximal operator as a gradient descent algorithm over the Moreau envelope. Using Fenchel's duality theorem, one can derive the following dual formulation of the Moreau envelope:
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 ...