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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 , ().
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 ...
Backpropagation computes the gradient of a loss function with respect to the weights of the network for a single input–output example, and does so efficiently, computing the gradient one layer at a time, iterating backward from the last layer to avoid redundant calculations of intermediate terms in the chain rule; this can be derived through ...
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]
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.
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]
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This includes, for example, early stopping, using a robust loss function, and discarding outliers. Implicit regularization is essentially ubiquitous in modern machine learning approaches, including stochastic gradient descent for training deep neural networks, and ensemble methods (such as random forests and gradient boosted trees).