[doc][hackathon] To add Adadelta Optimizer to the documentation (#63255)

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  #63236.

In this PR we are adding description of AdaDelta Algorithm to the documentation.  For more details, we refer to the paper  here https://arxiv.org/abs/1212.5701

<img width="654" alt="AdaDeltaalg" src="https://user-images.githubusercontent.com/73658284/132770544-82ccf90a-1d54-4ad5-8fc4-51c8dec63a12.png">

Pull Request resolved: #63255

Reviewed By: ngimel

Differential Revision: D30867589

Pulled By: iramazanli

fbshipit-source-id: 5ba602c20c724a4486bdd38b73e1b64c0e767bdc

torch/optim/adadelta.py

@@ -5,9 +5,33 @@
 
 
 class Adadelta(Optimizer):
-    """Implements Adadelta algorithm.
-
-    It has been proposed in `ADADELTA: An Adaptive Learning Rate Method`__.
+    r"""Implements Adadelta algorithm.
+
+    .. math::
+       \begin{aligned}
+            &\rule{110mm}{0.4pt}                                                                 \\
+            &\textbf{input}      : \gamma \text{ (lr)}, \: \theta_0 \text{ (params)},
+                \: f(\theta) \text{ (objective)}, \: \rho \text{ (decay)},
+                \: \lambda \text{ (weight decay)}                                                \\
+            &\textbf{initialize} :  v_0  \leftarrow 0 \: \text{ (square avg)},
+                \: u_0 \leftarrow 0 \: \text{ (accumulate variables)}                     \\[-1.ex]
+            &\rule{110mm}{0.4pt}                                                                 \\
+            &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do}                         \\
+            &\hspace{5mm}g_t           \leftarrow   \nabla_{\theta} f_t (\theta_{t-1})           \\
+            &\hspace{5mm}if \: \lambda \neq 0                                                    \\
+            &\hspace{10mm} g_t \leftarrow g_t + \lambda  \theta_{t-1}                            \\
+            &\hspace{5mm} v_t      \leftarrow v_{t-1} \rho + g^2_t (1 - \rho)                    \\
+            &\hspace{5mm}\Delta x_t    \leftarrow   \frac{\sqrt{u_{t-1} +
+                \epsilon }}{ \sqrt{v_t + \epsilon}  }g_t \hspace{21mm}                           \\
+            &\hspace{5mm} u_t  \leftarrow   u_{t-1}  \rho +
+                 \Delta x^2_t  (1 - \rho)                                                        \\
+            &\hspace{5mm}\theta_t      \leftarrow   \theta_{t-1} - \gamma  \Delta x_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 `ADADELTA: An Adaptive Learning Rate Method`_.
 
     Args:
         params (iterable): iterable of parameters to optimize or dicts defining
@@ -20,7 +44,8 @@ class Adadelta(Optimizer):
             to the parameters (default: 1.0)
         weight_decay (float, optional): weight decay (L2 penalty) (default: 0)
 
-    __ https://arxiv.org/abs/1212.5701
+    .. _ADADELTA\: An Adaptive Learning Rate Method:
+        https://arxiv.org/abs/1212.5701
     """
 
     def __init__(self, params, lr=1.0, rho=0.9, eps=1e-6, weight_decay=0):