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From tillrohrmann <...@git.apache.org>
Date Thu, 07 May 2015 09:48:13 GMT
Github user tillrohrmann commented on a diff in the pull request:

--- Diff: docs/libs/ml/optimization.md ---
@@ -0,0 +1,222 @@
+---
+mathjax: include
+title: "ML - Optimization"
+displayTitle: <a href="index.md">ML</a> - Optimization
+---
+<!--
+Licensed to the Apache Software Foundation (ASF) under one
+or more contributor license agreements.  See the NOTICE file
+distributed with this work for additional information
+to you under the Apache License, Version 2.0 (the
+"License"); you may not use this file except in compliance
+with the License.  You may obtain a copy of the License at
+
+
+Unless required by applicable law or agreed to in writing,
+"AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+KIND, either express or implied.  See the License for the
+specific language governing permissions and limitations
+-->
+
+{:toc}
+
+
+## Mathematical Formulation
+
+The optimization framework in Flink is a developer-oriented package that can be used
to solve
+[optimization](https://en.wikipedia.org/wiki/Mathematical_optimization)
+problems common in Machine Learning (ML) tasks. In the supervised learning context, this
usually
+involves finding a model, as defined by a set of parameters $w$, that minimize a function
$f(\wv)$
+given a set of $(\x, y)$ examples,
+where $\x$ is a feature vector and $y$ is a real number, which can represent either a
real value in
+the regression case, or a class label in the classification case. In supervised learning,
the
+function to be minimized is usually of the form:
+
+$$+$$+ f(\wv) := + \frac1n \sum_{i=1}^n L(\wv;\x_i,y_i) + + \lambda\, R(\wv) + \label{eq:objectiveFunc} + \ . +$$ +$$
+
+where $L$ is the loss function and $R(\wv)$ the regularization penalty. We use $L$ to
measure how
+well the model fits the observed data, and we use $R$ in order to impose a complexity
cost to the
+model, with $\lambda > 0$ being the regularization parameter.
+
+### Loss Functions
+
+In supervised learning, we use loss functions in order to measure the model fit, by
+penalizing errors in the predictions $p$ made by the model compared to the true $y$ for
each
+example. Different loss function can be used for regression (e.g. Squared Loss) and classification
+(e.g. Hinge Loss).
+
+Some common loss functions are:
+
+* Squared Loss: $\frac{1}{2} (\wv^T \x - y)^2, \quad y \in \R$
+* Hinge Loss: $\max (0, 1 - y ~ \wv^T \x), \quad y \in \{-1, +1\}$
+* Logistic Loss: $\log(1+\exp( -y ~ \wv^T \x)), \quad y \in \{-1, +1\}$
+
+Currently, only the Squared Loss function is implemented in Flink.
+
+### Regularization Types
+
+[Regularization](https://en.wikipedia.org/wiki/Regularization_(mathematics)) in machine
learning is
+imposes penalties to the estimated models, in order to reduce overfitting. The most common
penalties
+are the $L_1$ and $L_2$ penalties, defined as:
+
+* $L_1$: $R(\wv) = \|\wv\|_1$
+* $L_2$: $R(\wv) = \frac{1}{2}\|\wv\|_2^2$
+
+The $L_2$ penalty penalizes large weights, favoring solutions with more small weights
rather than
+few large ones.
+The $L_1$ penalty can be used to drive a number of the solution coefficients to 0, thereby
+producing sparse solutions.
+The optimization framework in Flink supports the $L_1$ and $L_2$ penalties, as well as
no
+regularization. The
+regularization parameter $\lambda$ in $\eqref{objectiveFunc}$ determines the amount of

+regularization applied to the model,
+and is usually determined through model cross-validation.
+
+
+In order to find a (local) minimum of a function, Gradient Descent methods take steps
in the
+direction opposite to the gradient of the function $\eqref{objectiveFunc}$ taken with
+respect to the current parameters (weights).
+In order to compute the exact gradient we need to perform one pass through all the points
in
+a dataset, making the process computationally expensive.
+An alternative is Stochastic Gradient Descent (SGD) where at each iteration we sample
one point
+from the complete dataset and update the parameters for each point, in an online manner.
+
+In mini-batch SGD we instead sample random subsets of the dataset, and compute the gradient
+over each batch. At each iteration of the algorithm we update the weights once, based
on
+the average of the gradients computed from each mini-batch.
+
+An important parameter is the learning rate $\eta$, or step size, which is currently
determined as
+$\eta = \frac{\eta_0}{\sqrt{j}}$, where $\eta_0$ is the initial step size and $j$ is
the iteration
+number. The setting of the initial step size can significantly affect the performance
of the
+algorithm. For some practical tips on tuning SGD see Leon Botou's
+
+The current implementation of SGD  uses the whole partition, making it
+effectively a batch gradient descent. Once a sampling operator has been introduced in
+mini-batch SGD will be performed.
+
+
+### Parameters
+
+  The stochastic gradient descent implementation can be controlled by the following parameters:
+
+   <table class="table table-bordered">
+      <tr>
+        <th class="text-left" style="width: 20%">Parameter</th>
+        <th class="text-center">Description</th>
+      </tr>
+    <tbody>
+      <tr>
+        <td><strong>Loss Function</strong></td>
+        <td>
+          <p>
+            The class of the loss function to be used. (Default value:
+            <strong>SquaredLoss</strong>, used for regression tasks)
+          </p>
+        </td>
+      </tr>
+      <tr>
+        <td><strong>RegularizationType</strong></td>
+        <td>
+          <p>
+            The type of regularization penalty to apply. (Default value:
+            <strong>NoRegularization</strong>)
+          </p>
+        </td>
+      </tr>
+      <tr>
+        <td><strong>RegularizationParameter</strong></td>
+        <td>
+          <p>
+            The amount of regularization to apply. (Default value:<strong>0</strong>)
+          </p>
+        </td>
+      </tr>
+      <tr>
+        <td><strong>Iterations</strong></td>
+        <td>
+          <p>
+            The maximum number of iterations. (Default value: <strong>10</strong>)
+          </p>
+        </td>
+      </tr>
+      <tr>
+        <td><strong>Stepsize</strong></td>
+        <td>
+          <p>
+            Initial step size for the gradient descent method.
+            This value controls how far the gradient descent method moves in the opposite
+            (Default value: <strong>0.1</strong>)
+          </p>
+        </td>
+      </tr>
+    </tbody>
+  </table>
--- End diff --

I think the PredictionFunction parameter is missing.

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