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Adaptive simulated annealing algorithms address this problem by connecting the cooling schedule to the search progress. Other adaptive approaches such as Thermodynamic Simulated Annealing, [16] automatically adjusts the temperature at each step based on the energy difference between the two states, according to the laws of thermodynamics.
The properties of gradient descent depend on the properties of the objective function and the variant of gradient descent used (for example, if a line search step is used). The assumptions made affect the convergence rate, and other properties, that can be proven for gradient descent. [ 33 ]
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
Adaptive simulated annealing (ASA) is a variant of simulated annealing (SA) algorithm in which the algorithm parameters that control temperature schedule and random step selection are automatically adjusted according to algorithm progress. This makes the algorithm more efficient and less sensitive to user defined parameters than canonical SA.
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 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.
If reduction of is rapid, a smaller value can be used, bringing the algorithm closer to the Gauss–Newton algorithm, whereas if an iteration gives insufficient reduction in the residual, can be increased, giving a step closer to the gradient-descent direction. Note that the gradient of with respect to equals ([()]).
SPSA is a descent method capable of finding global minima, sharing this property with other methods such as simulated annealing. Its main feature is the gradient approximation that requires only two measurements of the objective function, regardless of the dimension of the optimization problem.