Gradient descent with decaying learning rate

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Gradient descent with decaying learning rate is a form of gradient descent where the learning rate varies as a function of the number of iterations, but is not otherwise dependent on the value of the vector at the stage. The update rule is as follows:

\vec{x}^{(k+1)} = \vec{x}^{(k)} - \alpha_k f\left(\vec{x^{(k)}}\right)

where \alpha_k depends only on k and not on the choice of x^{(k)}.


Type of decay Example expression for \alpha_k More information
linear decay \alpha_k = \frac{\alpha_0}{k + 1} Gradient descent with linearly decaying learning rate
quadratic decay \alpha_k = \frac{\alpha_0}{(k + 1)^2} Gradient descent with quadratically decaying learning rate
exponential decay \alpha_k = \alpha_0 e^{-\beta k} where \beta > 0 Gradient descent with exponentially decaying learning rate