class AdaDelta(learning_rate: float | Callable[[array], array], rho: float = 0.9, eps: float = 1e-06)#

The AdaDelta optimizer with a learning rate [1].

Our AdaDelta implementation follows the original paper. In detail,

[1]: Zeiler, M.D., 2012. ADADELTA: an adaptive learning rate method. arXiv preprint arXiv:1212.5701.

\[\begin{split}v_{t+1} &= \rho v_t + (1 - \rho) g_t^2 \\ \Delta w_{t+1} &= \frac{\sqrt{u_t + \epsilon}}{\sqrt{v_{t+1} + \epsilon}} g_t \\ u_{t+1} &= \rho u_t + (1 - \rho) \Delta w_{t+1}^2 \\ w_{t+1} &= w_t - \lambda \Delta w_{t+1}\end{split}\]
  • learning_rate (float or callable) – The learning rate \(\lambda\).

  • rho (float, optional) – The coefficient \(\rho\) used for computing a running average of squared gradients. Default: 0.9

  • eps (float, optional) – The term \(\epsilon\) added to the denominator to improve numerical stability. Default: 1e-8


__init__(learning_rate[, rho, eps])

apply_single(gradient, parameter, state)

Performs the AdaDelta parameter update and stores \(v\) and \(u\) in the optimizer state.

init_single(parameter, state)

Initialize optimizer state