diff --git a/doc/modules/sgd.rst b/doc/modules/sgd.rst
index 2e202eb4ab0d771037297a65bf04b59e1904ebb3..d6f21ae9ea23309a424057e7988c79ff5e823846 100644
--- a/doc/modules/sgd.rst
+++ b/doc/modules/sgd.rst
@@ -7,50 +7,52 @@ Stochastic Gradient Descent
.. currentmodule:: scikits.learn.linear_model
-**Stochastic Gradient Descent (SGD)** is a simple yet very efficient approach
-to discriminative learning of linear classifiers under convex loss functions
-such as (linear) `Support Vector Machines <http://en.wikipedia.org/wiki/Support_vector_machine>`_ and `Logistic Regression <http://en.wikipedia.org/wiki/Logistic_regression>`_.
-Even though SGD has been around in the machine learning community for a long time,
-it has received a considerable amount of attention just recently in the
-context of large-scale learning.
-
-SGD has been successfully applied to large-scale and sparse machine learning
-problems often encountered in text classification and natural language
-processing.
-Given that the data is sparse, the classifiers in this module easily scale
-to problems with more than 10^5 training examples and more than 10^4 features.
+**Stochastic Gradient Descent (SGD)** is a simple yet very efficient
+approach to discriminative learning of linear classifiers under
+convex loss functions such as (linear) `Support Vector Machines
+<http://en.wikipedia.org/wiki/Support_vector_machine>`_ and `Logistic
+Regression <http://en.wikipedia.org/wiki/Logistic_regression>`_.
+Even though SGD has been around in the machine learning community for
+a long time, it has received a considerable amount of attention just
+recently in the context of large-scale learning.
+
+SGD has been successfully applied to large-scale and sparse machine
+learning problems often encountered in text classification and natural
+language processing. Given that the data is sparse, the classifiers
+in this module easily scale to problems with more than 10^5 training
+examples and more than 10^4 features.
The advantages of Stochastic Gradient Descent are:
- Efficiency.
- - Ease of implementation (lots of opportunities for code tuning).
+ - Ease of implementation (lots of opportunities for code tuning).
The disadvantages of Stochastic Gradient Descent include:
- SGD requires a number of hyperparameters including the regularization
- parameter and the number of iterations.
+ parameter and the number of iterations.
- - SGD is sensitive to feature scaling.
+ - SGD is sensitive to feature scaling.
Classification
==============
-.. warning:: Make sure you permute (shuffle) your training data before fitting the model or use `shuffle=True` to shuffle after each iterations.
+.. warning:: Make sure you permute (shuffle) your training data before fitting the model or use `shuffle=True` to shuffle after each iterations.
-The class :class:`SGDClassifier` implements a plain stochastic gradient descent
-learning routine which supports different loss functions and penalties for
-classification.
+The class :class:`SGDClassifier` implements a plain stochastic gradient
+descent learning routine which supports different loss functions and
+penalties for classification.
.. figure:: ../auto_examples/linear_model/images/plot_sgd_separating_hyperplane.png
:target: ../auto_examples/linear_model/plot_sgd_separating_hyperplane.html
:align: center
:scale: 75
-As other classifiers, SGD has to be fitted with two arrays:
-an array X of size [n_samples, n_features] holding the training
-samples, and an array Y of size [n_samples] holding the target values
-(class labels) for the training samples::
+As other classifiers, SGD has to be fitted with two arrays: an array X
+of size [n_samples, n_features] holding the training samples, and an
+array Y of size [n_samples] holding the target values (class labels)
+for the training samples::
>>> from scikits.learn.linear_model import SGDClassifier
>>> X = [[0., 0.], [1., 1.]]
@@ -65,8 +67,8 @@ After being fitted, the model can then be used to predict new values::
>>> clf.predict([[2., 2.]])
array([ 1.])
-SGD fits a linear model to the training data. The member `coef_` holds the
-model parameters:
+SGD fits a linear model to the training data. The member `coef_` holds
+the model parameters:
>>> clf.coef_
array([ 9.90090187, 9.90090187])
@@ -76,61 +78,65 @@ Member `intercept_` holds the intercept (aka offset or bias):
>>> clf.intercept_
array(-9.9900299301496904)
-Whether or not the model should use an intercept, i.e. a biased hyperplane, is
-controlled by the parameter `fit_intercept`.
+Whether or not the model should use an intercept, i.e. a biased
+hyperplane, is controlled by the parameter `fit_intercept`.
To get the signed distance to the hyperplane use `decision_function`:
>>> clf.decision_function([[2., 2.]])
array([ 29.61357756])
-The concrete loss function can be set via the `loss` parameter. :class:`SGDClassifier` supports the
-following loss functions:
+The concrete loss function can be set via the `loss`
+parameter. :class:`SGDClassifier` supports the following loss functions:
- `loss="hinge"`: (soft-margin) linear Support Vector Machine.
- - `loss="modified_huber"`: smoothed hinge loss.
+ - `loss="modified_huber"`: smoothed hinge loss.
- `loss="log"`: Logistic Regression
-The first two loss functions are lazy, they only update the model parameters if
-an example violates the margin constraint, which makes training very efficient.
-Log loss, on the other hand, provides probability estimates.
+The first two loss functions are lazy, they only update the model
+parameters if an example violates the margin constraint, which makes
+training very efficient. Log loss, on the other hand, provides
+probability estimates.
-In the case of binary classification and `loss="log"` you get a probability
-estimate P(y=C|x) using `predict_proba`, where `C` is the largest class label:
-
- >>> clf = SGDClassifier(loss="log").fit(X, y)
- >>> clf.predict_proba([[1., 1.]])
- array([ 0.99999949])
+In the case of binary classification and `loss="log"` you get a
+probability estimate P(y=C|x) using `predict_proba`, where `C` is the
+largest class label:
-The concrete penalty can be set via the `penalty` parameter. `SGD` supports the
-following penalties:
+ >>> clf = SGDClassifier(loss="log").fit(X, y) >>>
+ clf.predict_proba([[1., 1.]]) array([ 0.99999949])
+
+The concrete penalty can be set via the `penalty` parameter. `SGD`
+supports the following penalties:
- `penalty="l2"`: L2 norm penalty on `coef_`.
- `penalty="l1"`: L1 norm penalty on `coef_`.
- - `penalty="elasticnet"`: Convex combination of L2 and L1; `rho * L2 + (1 - rho) * L1`.
-
-The default setting is `penalty="l2"`. The L1 penalty leads to sparse solutions,
-driving most coefficients to zero. The Elastic Net solves some deficiencies of
-the L1 penalty in the presence of highly correlated attributes. The parameter `rho`
-has to be specified by the user.
-
-:class:`SGDClassifier` supports multi-class classification by combining multiple
-binary classifiers in a "one versus all" (OVA) scheme. For each of the `K` classes,
-a binary classifier is learned that discriminates between that and all other `K-1`
-classes. At testing time, we compute the confidence score (i.e. the signed distances
-to the hyperplane) for each classifier and choose the class with the highest
-confidence. The Figure below illustrates the OVA approach on the iris dataset.
-The dashed lines represent the three OVA classifiers;
-the background colors show the decision surface induced by the three classifiers.
+ - `penalty="elasticnet"`: Convex combination of L2 and L1; `rho * L2 + (1 - rho) * L1`.
+
+The default setting is `penalty="l2"`. The L1 penalty leads to sparse
+solutions, driving most coefficients to zero. The Elastic Net solves
+some deficiencies of the L1 penalty in the presence of highly correlated
+attributes. The parameter `rho` has to be specified by the user.
+
+:class:`SGDClassifier` supports multi-class classification by combining
+multiple binary classifiers in a "one versus all" (OVA) scheme. For each
+of the `K` classes, a binary classifier is learned that discriminates
+between that and all other `K-1` classes. At testing time, we compute the
+confidence score (i.e. the signed distances to the hyperplane) for each
+classifier and choose the class with the highest confidence. The Figure
+below illustrates the OVA approach on the iris dataset. The dashed
+lines represent the three OVA classifiers; the background colors show
+the decision surface induced by the three classifiers.
.. figure:: ../auto_examples/linear_model/images/plot_sgd_iris.png
:target: ../auto_examples/linear_model/plot_sgd_iris.html
:align: center
:scale: 75
-In the case of multi-class classification `coef_` is a two-dimensionaly array of shape
-[n_classes, n_features] and `intercept_` is a one dimensional array of shape [n_classes]. The i-th row of `coef_` holds the weight vector of the OVA classifier for the i-th
-class; classes are indexed in ascending order (see member `classes`).
+In the case of multi-class classification `coef_` is a two-dimensionaly
+array of shape [n_classes, n_features] and `intercept_` is a one
+dimensional array of shape [n_classes]. The i-th row of `coef_` holds
+the weight vector of the OVA classifier for the i-th class; classes are
+indexed in ascending order (see member `classes`).
.. topic:: Examples:
@@ -140,17 +146,17 @@ class; classes are indexed in ascending order (see member `classes`).
Regression
==========
-The class :class:`SGDRegressor` implements a plain stochastic gradient descent learning
-routine which supports different loss functions and penalties to fit linear regression
-models.
+The class :class:`SGDRegressor` implements a plain stochastic gradient
+descent learning routine which supports different loss functions and
+penalties to fit linear regression models.
.. figure:: ../auto_examples/linear_model/images/plot_sgd_ols.png
:target: ../auto_examples/linear_model/plot_sgd_ols.html
:align: center
:scale: 75
-The concrete loss function can be set via the `loss` parameter. :class:`SGDRegressor` supports the
-following loss functions:
+The concrete loss function can be set via the `loss`
+parameter. :class:`SGDRegressor` supports the following loss functions:
- `loss="squared_loss"`: Ordinary least squares.
- `loss="huber"`: Huber loss for robust regression.
@@ -165,7 +171,9 @@ following loss functions:
Stochastic Gradient Descent for sparse data
===========================================
-.. note:: The sparse implementation produces slightly different results than the dense implementation due to a shrunk learning rate for the intercept.
+.. note:: The sparse implementation produces slightly different results
+ than the dense implementation due to a shrunk learning rate for the
+ intercept.
There is support for sparse data given in any matrix in a format
supported by scipy.sparse. Classes have the same name, just prefixed
@@ -186,14 +194,14 @@ Implemented classes are :class:`SGDClassifier` and :class:`SGDRegressor`.
Complexity
==========
-The major advantage of SGD is its efficiency, which is basically
-linear in the number of training examples. If X is a matrix of size (n, p)
-training has a cost of :math:`O(k n \bar p)`, where k is the number
-of iterations (epochs) and :math:`\bar p` is the average number of
-non-zero attributes per sample.
+The major advantage of SGD is its efficiency, which is basically
+linear in the number of training examples. If X is a matrix of size (n, p)
+training has a cost of :math:`O(k n \bar p)`, where k is the number
+of iterations (epochs) and :math:`\bar p` is the average number of
+non-zero attributes per sample.
-Recent theoretical results, however, show that the runtime to get some
-desired optimization accuracy does not increase as the training set size increases.
+Recent theoretical results, however, show that the runtime to get some
+desired optimization accuracy does not increase as the training set size increases.
Tips on Practical Use
=====================
@@ -206,51 +214,52 @@ Tips on Practical Use
results. See `The CookBook
<https://sourceforge.net/apps/trac/scikit-learn/wiki/CookBook>`_
for some examples on scaling. If your attributes have an intrinsic
- scale (e.g. word frequencies or indicator features) scaling is
- not needed.
+ scale (e.g. word frequencies or indicator features) scaling is
+ not needed.
- * Finding a reasonable regularization term :math:`\alpha` is
+ * Finding a reasonable regularization term :math:`\alpha` is
best done using grid search `for alpha in 10.0**-np.arange(1,7)`.
- * Empirically, we found that SGD converges after observing
- approx. 10^6 training samples. Thus, a reasonable first guess
- for the number of iterations is `n_iter = np.ceil(10**6 / n)`,
+ * Empirically, we found that SGD converges after observing
+ approx. 10^6 training samples. Thus, a reasonable first guess
+ for the number of iterations is `n_iter = np.ceil(10**6 / n)`,
where `n` is the size of the training set.
.. topic:: References:
- * `"Efficient BackProp" <yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf>`_
- Y. LeCun, L. Bottou, G. Orr, K. Müller - In Neural Networks: Tricks of the Trade 1998.
+ * `"Efficient BackProp" <yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf>`_
+ Y. LeCun, L. Bottou, G. Orr, K. Müller - In Neural Networks: Tricks
+ of the Trade 1998.
.. _sgd_mathematical_formulation:
Mathematical formulation
========================
-Given a set of training examples :math:`(x_1, y_1), \ldots, (x_n, y_n)` where
-:math:`x_i \in \mathbf{R}^n` and :math:`y_i \in \{-1,1\}`, our goal is to
+Given a set of training examples :math:`(x_1, y_1), \ldots, (x_n, y_n)` where
+:math:`x_i \in \mathbf{R}^n` and :math:`y_i \in \{-1,1\}`, our goal is to
learn a linear scoring function :math:`f(x) = w^T x + b` with model parameters
:math:`w \in \mathbf{R}^m` and intercept :math:`b \in \mathbf{R}`. In order
to make predictions, we simply look at the sign of :math:`f(x)`.
-A common choice to find the model parameters is by minimizing the regularized
+A common choice to find the model parameters is by minimizing the regularized
training error given by
.. math::
E(w,b) = \sum_{i=1}^{n} L(y_i, f(x_i)) + \alpha R(w)
-where :math:`L` is a loss function that measures model (mis)fit and :math:`R` is a
-regularization term (aka penalty) that penalizes model complexity; :math:`\alpha > 0`
-is a non-negative hyperparameter.
+where :math:`L` is a loss function that measures model (mis)fit and
+:math:`R` is a regularization term (aka penalty) that penalizes model
+complexity; :math:`\alpha > 0` is a non-negative hyperparameter.
-Different choices for :math:`L` entail different classifiers such as
+Different choices for :math:`L` entail different classifiers such as
- Hinge: (soft-margin) Support Vector Machines.
- Log: Logistic Regression.
- - Least-Squares: Ridge Regression.
+ - Least-Squares: Ridge Regression.
-All of the above loss functions can be regarded as an upper bound on the
-misclassification error (Zero-one loss) as shown in the Figure below.
+All of the above loss functions can be regarded as an upper bound on the
+misclassification error (Zero-one loss) as shown in the Figure below.
.. figure:: ../auto_examples/linear_model/images/plot_sgd_loss_functions.png
:align: center
@@ -258,12 +267,13 @@ misclassification error (Zero-one loss) as shown in the Figure below.
Popular choices for the regularization term :math:`R` include:
- - L2 norm: :math:`R(w) := \frac{1}{2} \sum_{i=1}^{n} w_i^2`,
- - L1 norm: :math:`R(w) := \sum_{i=1}^{n} |w_i|`, which leadsin sparse solutions.
- - Elastic Net: :math:`R(w) := \rho \frac{1}{2} \sum_{i=1}^{n} w_i^2 + (1-\rho) \sum_{i=1}^{n} |w_i|`, a convex combination of L2 and L1.
+ - L2 norm: :math:`R(w) := \frac{1}{2} \sum_{i=1}^{n} w_i^2`,
+ - L1 norm: :math:`R(w) := \sum_{i=1}^{n} |w_i|`, which leads to sparse
+ solutions.
+ - Elastic Net: :math:`R(w) := \rho \frac{1}{2} \sum_{i=1}^{n} w_i^2 + (1-\rho) \sum_{i=1}^{n} |w_i|`, a convex combination of L2 and L1.
-The Figure below shows the contours of the different regularization terms
-in the parameter space when :math:`R(w) = 1`.
+The Figure below shows the contours of the different regularization terms
+in the parameter space when :math:`R(w) = 1`.
.. figure:: ../auto_examples/linear_model/images/plot_sgd_penalties.png
:align: center
@@ -272,25 +282,26 @@ in the parameter space when :math:`R(w) = 1`.
SGD
---
-Stochastic gradient descent is an optimization method for unconstrained
+Stochastic gradient descent is an optimization method for unconstrained
optimization problems. In contrast to (batch) gradient descent, SGD
-approximates the true gradient of :math:`E(w,b)` by considering a
-single training example at a time.
+approximates the true gradient of :math:`E(w,b)` by considering a
+single training example at a time.
-The class :class:`SGDClassifier` implements a first-order SGD learning routine.
-The algorithm iterates over the training examples and for each example
-updates the model parameters according to the update rule given by
+The class :class:`SGDClassifier` implements a first-order SGD learning
+routine. The algorithm iterates over the training examples and for each
+example updates the model parameters according to the update rule given by
.. math::
- w \leftarrow w - \eta (\alpha \frac{\partial R(w)}{\partial w}
+ w \leftarrow w - \eta (\alpha \frac{\partial R(w)}{\partial w}
+ \frac{\partial L(w^T x_i + b, y_i)}{\partial w})
-where :math:`\eta` is the learning rate which controls the step-size
-in the parameter space.
-The intercept :math:`b` is updated similarly but without regularization.
+where :math:`\eta` is the learning rate which controls the step-size in
+the parameter space. The intercept :math:`b` is updated similarly but
+without regularization.
-The model parameters can be accessed through the members coef\_ and intercept\_:
+The model parameters can be accessed through the members coef\_ and
+intercept\_:
- Member coef\_ holds the weights :math:`w`
@@ -298,37 +309,39 @@ The model parameters can be accessed through the members coef\_ and intercept\_:
.. topic:: References:
- * `"Solving large scale linear prediction problems using stochastic gradient descent algorithms"
- <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.58.7377>`_
+ * `"Solving large scale linear prediction problems using stochastic
+ gradient descent algorithms"
+ <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.58.7377>`_
T. Zhang - In Proceedings of ICML '04.
-
+
* `"Regularization and variable selection via the elastic net"
- <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.124.4696>`_
- H. Zou, T. Hastie - Journal of the Royal Statistical Society Series B, 67 (2), 301-320.
+ <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.124.4696>`_
+ H. Zou, T. Hastie - Journal of the Royal Statistical Society Series B,
+ 67 (2), 301-320.
Implementation details
======================
-The implementation of SGD is influenced by the `Stochastic Gradient SVM
-<http://leon.bottou.org/projects/sgd>`_ of Léon Bottou. Similar to SvmSGD,
-the weight vector is represented as the product of a scalar and a vector
-which allows an efficient weight update in the case of L2 regularization.
-In the case of sparse feature vectors, the intercept is updated with a
-smaller learning rate (multiplied by 0.01) to account for the fact that
+The implementation of SGD is influenced by the `Stochastic Gradient SVM
+<http://leon.bottou.org/projects/sgd>`_ of Léon Bottou. Similar to SvmSGD,
+the weight vector is represented as the product of a scalar and a vector
+which allows an efficient weight update in the case of L2 regularization.
+In the case of sparse feature vectors, the intercept is updated with a
+smaller learning rate (multiplied by 0.01) to account for the fact that
it is updated more frequently. Training examples are picked up sequentially
-and the learning rate is lowered after each observed example. We adopted the
-learning rate schedule from Shalev-Shwartz et al. 2007.
-For multi-class classification, a "one versus all" approach is used.
-We use the truncated gradient algorithm proposed by Tsuruoka et al. 2009
-for L1 regularization (and the Elastic Net).
+and the learning rate is lowered after each observed example. We adopted the
+learning rate schedule from Shalev-Shwartz et al. 2007.
+For multi-class classification, a "one versus all" approach is used.
+We use the truncated gradient algorithm proposed by Tsuruoka et al. 2009
+for L1 regularization (and the Elastic Net).
The code is written in Cython.
.. topic:: References:
* `"Stochastic Gradient Descent" <http://leon.bottou.org/projects/sgd>`_ L. Bottou - Website, 2010.
- * `"Pegasos: Primal estimated sub-gradient solver for svm"
+ * `"Pegasos: Primal estimated sub-gradient solver for svm"
<http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.74.8513>`_
S. Shalev-Shwartz, Y. Singer, N. Srebro - In Proceedings of ICML '07.