deploy: a82e3e9

.buildinfo

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+# Sphinx build info version 1
+# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
+config: 02ef5ebb17889f7b71c7616990e0868f
+tags: 645f666f9bcd5a90fca523b33c5a78b7

_sources/gradientdescent.rst.txt

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+.. _gradientdescent:
+
+Gradient Descent
+================
+
+What is gradient descent?
+-------------------------
+
+Gradient descent is an iterative optimization algorithm that can be used to find a minimum of a function :math:`f(\theta)`. At each iteration, it takes a step in the direction of the *negative* gradient :math:`\nabla f(\theta)`. The algorithm can also be used to find a maximum. In that case, the algorithm should take a step in the direction of the gradient.
+
+How does gradient descent work?
+-------------------------------
+
+The gradient descent algorithm is as follows:
+
+.. topic:: Gradient Descent Algorithm
+
+	**Input**: Choose starting value :math:`\theta_0`, learning rate :math:`\epsilon` and take :math:`i=0`.
+
+	**Algorithm**:
+
+	1. Calculate :math:`\nabla f(\theta_i)`.
+	2. Update :math:`\theta_{i+1}=\theta_i-\epsilon \nabla f(\theta_i)`.
+	3. (a) stop if :math:`i` large or :math:`|\theta_{i+1}-\theta_{i}|` small enough.
+	   (b) else: update :math:`i` to :math:`i+1` and go back to (1).
+
+
+
+Binder
+------
+
+If you want to experiment with the gradient descent algorithm, you can use the `provided Jupyter Notebook <https://github.com/NanneD/SOLT/blob/main/notebooks/GradientDescent.ipynb>`_. You can run the notebook directly in your browser by using Binder; simply click on the following button to open the notebook:
+
+.. image:: https://mybinder.org/badge_logo.svg
+ :target: https://mybinder.org/v2/gh/NanneD/SOLT/HEAD?labpath=notebooks%2FGradientDescent.ipynb

_sources/index.rst.txt

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+:html_theme.sidebar_secondary.remove:
+
+SOLT
+====
+
+| **Release** |release|
+| **Date** |today|
+
+SOLT (Stochastic Optimization Learning Tool) is a learning tool about gradient descent and stochastic optimization, and in particular the simultaneous perturbation stochastic approximation (SPSA) algorithm. This website contains information about gradient descent, SPSA and N-dimensional SPSA.
+
+You can experiment with the various algorithms through Jupyter Notebooks that contain implementations of the algorithms. You can directly open the notebooks in a browser with Binder by clicking on the following button:
+
+.. image:: https://mybinder.org/badge_logo.svg
+    :align: center
+    :target: https://mybinder.org/v2/gh/NanneD/SOLT/HEAD
+
+| **License**
+| The GNU Affero General Public License v3 (AGPL-3.0) license is used. For more information, please see the `GitHub repository <https://github.com/NanneD/SOLT>`_.
+
+| **Contributing**
+| Please see the README file on the `GitHub repository <https://github.com/NanneD/SOLT>`_ for information on how to contribute.
+
+.. grid:: 1 2 2 2
+
+    .. grid-item-card::
+	:link: gradientdescent
+        :width: 75%
+        :link-type: ref
+        :link-alt: Gradient Descent
+        :img-background: _static/GD_logo.svg
+
+    .. grid-item-card::
+        :link: spsa
+        :width: 75%
+        :link-type: ref
+        :link-alt: SPSA
+        :img-background: _static/SPSA_logo.svg
+
+
+.. toctree::
+   :hidden:
+
+   gradientdescent
+   spsa

_sources/spsa.rst.txt

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+.. _spsa:
+
+SPSA
+====
+
+What is SPSA?
+-------------
+
+Simultaneous Perturbation Stochastic Approximation (SPSA) is an algorithm developed by `Spall <https://www.jhuapl.edu/spsa/>`_. It can be used if noisy and unbiased measurements of the gradient :math:`g(\boldsymbol{\theta}`) are available. It can also be used if only (noisy) measurements of the loss function :math:`f(\boldsymbol{\theta})` are available.
+
+The advantage of SPSA compared to other algorithms is that only two loss measurements are required to generate an update. Therefore, SPSA is scalable.
+
+How does SPSA work?
+-------------------
+
+Let :math:`\eta_i \in (0, \infty)` be a perturbation vector and :math:`\Delta_i` be a random vector such that :math:`\{\Delta_i\}` is an iid sequence with :math:`\Delta_i(k)` and :math:`1/\Delta_i(k)` bounded and symmetric around zero. The components :math:`\Delta_i(k)` are mutually independent. In practice, often the following binary random variable is used for :math:`\Delta_i`:
+
+.. math::
+
+	\mathbb{P}(\Delta_i(j) = -1) = \frac{1}{2} = \mathbb{P}(\Delta_i(j) = 1),
+
+for all :math:`i` and :math:`j`.
+
+
+The SPSA algorithm is then as follows:
+
+.. topic:: SPSA Algorithm
+
+	**Input**: Choose starting value :math:`\theta_0`, learning rate :math:`\epsilon` and take :math:`i = 0`.
+
+	**Algorithm**:
+
+	1. Calculate :math:`(g_i^{SPSA}(\theta_i))(j) = \frac{f(\theta_i+\eta_i\Delta_i)-f(\theta_i-\eta_i\Delta_i)}{2\eta_i\Delta_i(j)}`. 
+	2. Update :math:`\theta_{i+1} = \theta_{i} - \epsilon (g_i^{SPSA}(\theta_i))`.
+	3. (a) stop if :math:`i` large or :math:`|\theta_{i+1}-\theta_{i}|` small enough.  
+   	   (b) else: update :math:`i` to :math:`i+1` and go back to (1).
+
+Binder
+------
+
+If you want to experiment with the SPSA algorithm, you can use the `provided Jupyter Notebook <https://github.com/NanneD/SOLT/blob/main/notebooks/SPSA.ipynb>`_. If you want to experiment with the N-dimensional version of the algorithm, then you can use `this Jupyter Notebook <https://github.com/NanneD/SOLT/blob/main/notebooks/SPSA-ND.ipynb>`_.
+
+You can also run the notebooks directly in your browser by using Binder; simply click on the following button to open the SPSA notebook:
+
+.. image:: https://mybinder.org/badge_logo.svg
+ :target: https://mybinder.org/v2/gh/NanneD/SOLT/HEAD?labpath=notebooks%2FSPSA.ipynb

_sphinx_design_static/design-tabs.js

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+}
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