To add Rectified Adam Description to Documentation (#63772)

Summary:
It has been discussed before that adding description of Optimization algorithms to PyTorch Core documentation may result in a nice Optimization research tutorial. In the following tracking issue we mentioned about all the necessary algorithms and links to the originally published paper  https://github.com/pytorch/pytorch/issues/63236.

In this PR we are adding description of Rectified Adam Algorithm to the documentation.  For more details, we refer to the paper  https://arxiv.org/abs/1908.03265

<img width="446" alt="RadamAlgo" src="https://user-images.githubusercontent.com/73658284/132587815-4764b642-df53-4e41-975f-72e0f40fdc48.png">

Pull Request resolved: https://github.com/pytorch/pytorch/pull/63772

Reviewed By: datumbox

Differential Revision: D30839694

Pulled By: iramazanli

fbshipit-source-id: 6f5629ce56e10c66a451433334b587b99eda1610

diff --git a/torch/optim/radam.py b/torch/optim/radam.py
index 17636b6..63ad94c 100644 (file)
--- a/torch/optim/radam.py
+++ b/torch/optim/radam.py
@@ -4,9 +4,41 @@ from .optimizer import Optimizer
 
 
 class RAdam(Optimizer):
-    r"""implements RAdam algorithm.
+    r"""Implements RAdam algorithm.
 
-    It has been proposed in `On the variance of the adaptive learning rate and beyond`_.
+    .. math::
+       \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)}, \:
+                \lambda \text{ (weightdecay)},                                                   \\
+            &\hspace{13mm} \epsilon \text{ (epsilon)}                                            \\
+            &\textbf{initialize} :  m_0 \leftarrow 0 \text{ ( first moment)},
+                v_0 \leftarrow 0 \text{ ( second moment)},                                       \\
+            &\hspace{18mm} \rho_{\infty} \leftarrow 2/(1-\beta_2) -1                      \\[-1.ex]
+            &\rule{110mm}{0.4pt}  \\
+            &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do}                         \\
+            &\hspace{6mm}g_t           \leftarrow   \nabla_{\theta} f_t (\theta_{t-1})           \\
+            &\hspace{5mm} \textbf{if} \: \lambda \neq 0                                          \\
+            &\hspace{10mm} g_t \leftarrow g_t + \lambda \theta_{t-1}                             \\
+            &\hspace{6mm}m_t           \leftarrow   \beta_1 m_{t-1} + (1 - \beta_1) g_t          \\
+            &\hspace{6mm}v_t           \leftarrow   \beta_2 v_{t-1} + (1-\beta_2) g^2_t          \\
+            &\hspace{6mm}\widehat{m_t} \leftarrow   m_t/\big(1-\beta_1^t \big)                   \\
+            &\hspace{6mm}\rho_t \leftarrow \rho_{\infty} -
+                2 t \beta^t_2 /\big(1-\beta_2^t \big)                                    \\[0.1.ex]
+            &\hspace{6mm}\textbf{if} \: \rho_t > 5                                               \\
+            &\hspace{12mm} l_t \leftarrow \sqrt{ (1-\beta^t_2) / \big( v_t +\epsilon \big) }     \\
+            &\hspace{12mm} r_t \leftarrow
+      \sqrt{\frac{(\rho_t-4)(\rho_t-2)\rho_{\infty}}{(\rho_{\infty}-4)(\rho_{\infty}-2) \rho_t}} \\
+            &\hspace{12mm}\theta_t \leftarrow \theta_{t-1} - \gamma \widehat{m_t} r_t l_t        \\
+            &\hspace{6mm}\textbf{else}                                                           \\
+            &\hspace{12mm}\theta_t \leftarrow \theta_{t-1} - \gamma \widehat{m_t}                \\
+            &\rule{110mm}{0.4pt}                                                          \\[-1.ex]
+            &\bf{return} \:  \theta_t                                                     \\[-1.ex]
+            &\rule{110mm}{0.4pt}                                                          \\[-1.ex]
+       \end{aligned}
+
+    For further details regarding the algorithm we refer to `On the variance of the adaptive learning rate and beyond`_.
 
     Args:
         params (iterable): iterable of parameters to optimize or dicts defining