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It is particularly useful in machine learning for minimizing the cost or loss function. [1] Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. [2]
Consequently, the hinge loss function cannot be used with gradient descent methods or stochastic gradient descent methods which rely on differentiability over the entire domain. However, the hinge loss does have a subgradient at y f ( x → ) = 1 {\displaystyle yf({\vec {x}})=1} , which allows for the utilization of subgradient descent methods ...
In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. [4]
Clearly, =. giving us our final equation for the gradient: = ′ () As noted above, gradient descent tells us that our change for each weight should be proportional to the gradient.
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.
Like the related Davidon–Fletcher–Powell method, BFGS determines the descent direction by preconditioning the gradient with curvature information. It does so by gradually improving an approximation to the Hessian matrix of the loss function , obtained only from gradient evaluations (or approximate gradient evaluations) via a generalized ...
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.
The idea is to apply a steepest descent step to this minimization problem (functional gradient descent). The basic idea is to find a local minimum of the loss function by iterating on (). In fact, the local maximum-descent direction of the loss function is the negative gradient. [8]