AdamW

class brainpy.optim.AdamW(lr, train_vars=None, beta1=0.9, beta2=0.999, eps=1e-08, weight_decay=0.01, amsgrad=False, name=None)

Adam with weight decay regularization [1]_.

AdamW uses weight decay to regularize learning towards small weights, as this leads to better generalization. In SGD you can also use L2 regularization to implement this as an additive loss term, however L2 regularization does not behave as intended for adaptive gradient algorithms such as Adam.

\[ \begin{align}\begin{aligned}\begin{split}\begin{aligned} &\rule{110mm}{0.4pt} \\ &\textbf{input} : \gamma \text{(lr)}, \: \beta_1, \beta_2 \text{(betas)}, \: \theta_0 \text{(params)}, \: f(\theta) \text{(objective)}, \: \epsilon \text{ (epsilon)} \\ &\hspace{13mm} \lambda \text{(weight decay)}, \: \textit{amsgrad}, \: \textit{maximize} \\ &\textbf{initialize} : m_0 \leftarrow 0 \text{ (first moment)}, v_0 \leftarrow 0 \text{ ( second moment)}, \: \widehat{v_0}^{max}\leftarrow 0 \\[-1.ex] &\rule{110mm}{0.4pt} \\ &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do} \\\end{split}\\\begin{split} &\hspace{5mm}\textbf{if} \: \textit{maximize}: \\ &\hspace{10mm}g_t \leftarrow -\nabla_{\theta} f_t (\theta_{t-1}) \\ &\hspace{5mm}\textbf{else} \\ &\hspace{10mm}g_t \leftarrow \nabla_{\theta} f_t (\theta_{t-1}) \\ &\hspace{5mm} \theta_t \leftarrow \theta_{t-1} - \gamma \lambda \theta_{t-1} \\ &\hspace{5mm}m_t \leftarrow \beta_1 m_{t-1} + (1 - \beta_1) g_t \\ &\hspace{5mm}v_t \leftarrow \beta_2 v_{t-1} + (1-\beta_2) g^2_t \\ &\hspace{5mm}\widehat{m_t} \leftarrow m_t/\big(1-\beta_1^t \big) \\ &\hspace{5mm}\widehat{v_t} \leftarrow v_t/\big(1-\beta_2^t \big) \\ &\hspace{5mm}\textbf{if} \: amsgrad \\ &\hspace{10mm}\widehat{v_t}^{max} \leftarrow \mathrm{max}(\widehat{v_t}^{max}, \widehat{v_t}) \\ &\hspace{10mm}\theta_t \leftarrow \theta_t - \gamma \widehat{m_t}/ \big(\sqrt{\widehat{v_t}^{max}} + \epsilon \big) \\ &\hspace{5mm}\textbf{else} \\ &\hspace{10mm}\theta_t \leftarrow \theta_t - \gamma \widehat{m_t}/ \big(\sqrt{\widehat{v_t}} + \epsilon \big) \\ &\rule{110mm}{0.4pt} \\[-1.ex] &\bf{return} \: \theta_t \\[-1.ex] &\rule{110mm}{0.4pt} \\[-1.ex] \end{aligned}\end{split}\end{aligned}\end{align} \]

Parameters:

lr: float, Scheduler

learning rate.

beta1: optional, float

A positive scalar value for beta_1, the exponential decay rate for the first moment estimates. Generally close to 1.

beta2: optional, float

A positive scalar value for beta_2, the exponential decay rate for the second moment estimates. Generally close to 1.

eps: optional, float

A positive scalar value for epsilon, a small constant for numerical stability.

weight_decay: float

Strength of the weight decay regularization. Note that this weight decay is multiplied with the learning rate.

amsgrad: bool

whether to use the AMSGrad variant of this algorithm from the paper On the Convergence of Adam and Beyond.

nameoptional, str

The optimizer name.

References:

.. [1] Loshchilov, Ilya and Frank Hutter. “Decoupled Weight Decay Regularization.” International Conference on Learning Representations (2019).