mlx.optimizers.Adam#

class Adam(learning_rate: float | Callable[[array], array], betas: List[float] = [0.9, 0.999], eps: float = 1e-08)#

The Adam optimizer [1].

Our Adam implementation follows the original paper and omits the bias correction in the first and second moment estimates. In detail,

[1]: Kingma, D.P. and Ba, J., 2015. Adam: A method for stochastic optimization. ICLR 2015.

\[\begin{split}m_{t+1} &= \beta_1 m_t + (1 - \beta_1) g_t \\ v_{t+1} &= \beta_2 v_t + (1 - \beta_2) g_t^2 \\ w_{t+1} &= w_t - \lambda \frac{m_{t+1}}{\sqrt{v_{t+1} + \epsilon}}\end{split}\]
Parameters:

  • learning_rate (float or callable) – The learning rate \(\lambda\).

  • betas (Tuple[float, float], optional) – The coefficients \((\beta_1, \beta_2)\) used for computing running averages of the gradient and its square. Default: (0.9, 0.999)

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

Methods

__init__(learning_rate[, betas, eps])

apply_single(gradient, parameter, state)

Performs the Adam parameter update and stores \(v\) and \(m\) in the optimizer state.

init_single(parameter, state)

Initialize optimizer state