Github user dbtsai commented on a diff in the pull request:
https://github.com/apache/spark/pull/13796#discussion_r75215584
--- Diff:
mllib/src/main/scala/org/apache/spark/ml/classification/LogisticRegression.scala
---
@@ -933,32 +946,312 @@ class BinaryLogisticRegressionSummary
private[classification] (
}
/**
- * LogisticAggregator computes the gradient and loss for binary logistic
loss function, as used
- * in binary classification for instances in sparse or dense vector in an
online fashion.
- *
- * Note that multinomial logistic loss is not supported yet!
+ * LogisticAggregator computes the gradient and loss for binary or
multinomial logistic (softmax)
+ * loss function, as used in classification for instances in sparse or
dense vector in an online
+ * fashion.
*
- * Two LogisticAggregator can be merged together to have a summary of loss
and gradient of
+ * Two LogisticAggregators can be merged together to have a summary of
loss and gradient of
* the corresponding joint dataset.
*
+ * For improving the convergence rate during the optimization process and
also to prevent against
+ * features with very large variances exerting an overly large influence
during model training,
+ * packages like R's GLMNET perform the scaling to unit variance and
remove the mean in order to
+ * reduce the condition number. The model is then trained in this scaled
space, but returns the
+ * coefficients in the original scale. See page 9 in
+ * http://cran.r-project.org/web/packages/glmnet/glmnet.pdf
+ *
+ * However, we don't want to apply the
[[org.apache.spark.ml.feature.StandardScaler]] on the
+ * training dataset, and then cache the standardized dataset since it will
create a lot of overhead.
+ * As a result, we perform the scaling implicitly when we compute the
objective function (though
+ * we do not subtract the mean).
+ *
+ * Note that there is a difference between multinomial (softmax) and
binary loss. The binary case
+ * uses one outcome class as a "pivot" and regresses the other class
against the pivot. In the
+ * multinomial case, the softmax loss function is used to model each class
probability
+ * independently. Using softmax loss produces `K` sets of coefficients,
while using a pivot class
+ * produces `K - 1` sets of coefficients (a single coefficient vector in
the binary case). In the
+ * binary case, we can say that the coefficients are shared between the
positive and negative
+ * classes. When regularization is applied, multinomial (softmax) loss
will produce a result
+ * different from binary loss since the positive and negative don't share
the coefficients while the
+ * binary regression shares the coefficients between positive and negative.
+ *
+ * The following is a mathematical derivation for the multinomial
(softmax) loss.
+ *
+ * The probability of the multinomial outcome $y$ taking on any of the K
possible outcomes is:
+ *
+ * <p><blockquote>
+ * $$
+ * P(y_i=0|\vec{x}_i, \beta) = \frac{e^{\vec{x}_i^T
\vec{\beta}_0}}{\sum_{k=0}^{K-1}
+ * e^{\vec{x}_i^T \vec{\beta}_k}} \\
+ * P(y_i=1|\vec{x}_i, \beta) = \frac{e^{\vec{x}_i^T
\vec{\beta}_1}}{\sum_{k=0}^{K-1}
+ * e^{\vec{x}_i^T \vec{\beta}_k}}\\
+ * P(y_i=K-1|\vec{x}_i, \beta) = \frac{e^{\vec{x}_i^T
\vec{\beta}_{K-1}}\,}{\sum_{k=0}^{K-1}
+ * e^{\vec{x}_i^T \vec{\beta}_k}}
+ * $$
+ * </blockquote></p>
+ *
+ * The model coefficients $\beta = (\beta_1, \beta_2, ..., \beta_{K-1})$
become a matrix
+ * which has dimension of $K \times (N+1)$ if the intercepts are added. If
the intercepts are not
+ * added, the dimension will be $K \times N$.
+ *
+ * Note that the coefficients in the model above lack identifiability.
That is, any constant scalar
+ * can be added to all of the coefficients and the probabilities remain
the same.
+ *
+ * <p><blockquote>
+ * $$
+ * \begin{align}
+ * \frac{e^{\vec{x}_i^T \left(\vec{\beta}_0 +
\vec{c}\right)}}{\sum_{k=0}^{K-1}
+ * e^{\vec{x}_i^T \left(\vec{\beta}_k + \vec{c}\right)}}
+ * = \frac{e^{\vec{x}_i^T \vec{\beta}_0}e^{\vec{x}_i^T
\vec{c}}\,}{e^{\vec{x}_i^T \vec{c}}
+ * \sum_{k=0}^{K-1} e^{\vec{x}_i^T \vec{\beta}_k}}
+ * = \frac{e^{\vec{x}_i^T \vec{\beta}_0}}{\sum_{k=0}^{K-1}
e^{\vec{x}_i^T \vec{\beta}_k}}
+ * \end{align}
+ * $$
+ * </blockquote></p>
+ *
+ * However, when regularization is added to the loss function, the
coefficients are indeed
+ * identifiable because there is only one set of coefficients which
minimizes the regularization
+ * term. When no regularization is applied, we choose the coefficients
with the minimum L2
+ * penalty for consistency and reproducibility. For further discussion see:
+ *
+ * Friedman, et al. "Regularization Paths for Generalized Linear Models
via Coordinate Descent"
+ *
+ * The loss of objective function for a single instance of data (we do not
include the
+ * regularization term here for simplicity) can be written as
+ *
+ * <p><blockquote>
+ * $$
+ * \begin{align}
+ * \ell\left(\beta, x_i\right) &= -log{P\left(y_i \middle| \vec{x}_i,
\beta\right)} \\
+ * &= log\left(\sum_{k=0}^{K-1}e^{\vec{x}_i^T \vec{\beta}_k}\right) -
\vec{x}_i^T \vec{\beta}_y\\
+ * &= log\left(\sum_{k=0}^{K-1} e^{margins_k}\right) - margins_y
+ * \end{align}
+ * $$
+ * </blockquote></p>
+ *
+ * where ${margins}_k = \vec{x}_i^T \vec{\beta}_k$.
+ *
+ * For optimization, we have to calculate the first derivative of the loss
function, and a simple
+ * calculation shows that
+ *
+ * <p><blockquote>
+ * $$
+ * \begin{align}
+ * \frac{\partial \ell(\beta, \vec{x}_i, w_i)}{\partial \beta_{j, k}}
+ * &= x_{i,j} \cdot w_i \cdot \left(\frac{e^{\vec{x}_i \cdot
\vec{\beta}_k}}{\sum_{k'=0}^{K-1}
+ * e^{\vec{x}_i \cdot \vec{\beta}_{k'}}\,} - I_{y=k}\right) \\
+ * &= x_{i, j} \cdot w_i \cdot multiplier_k
+ * \end{align}
+ * $$
+ * </blockquote></p>
+ *
+ * where $w_i$ is the sample weight, $I_{y=k}$ is an indicator function
+ *
+ * <p><blockquote>
+ * $$
+ * I_{y=k} = \begin{cases}
+ * 1 & y = k \\
+ * 0 & else
+ * \end{cases}
+ * $$
+ * </blockquote></p>
+ *
+ * and
+ *
+ * <p><blockquote>
+ * $$
+ * multiplier_k = \left(\frac{e^{\vec{x}_i \cdot
\vec{\beta}_k}}{\sum_{k=0}^{K-1}
+ * e^{\vec{x}_i \cdot \vec{\beta}_k}} - I_{y=k}\right)
+ * $$
+ * </blockquote></p>
+ *
+ * If any of margins is larger than 709.78, the numerical computation of
multiplier and loss
+ * function will suffer from arithmetic overflow. This issue occurs when
there are outliers in
+ * data which are far away from the hyperplane, and this will cause the
failing of training once
+ * infinity is introduced. Note that this is only a concern when
max(margins) > 0.
+ *
+ * Fortunately, when max(margins) = maxMargin > 0, the loss function and
the multiplier can easily
+ * be rewritten into the following equivalent numerically stable formula.
+ *
+ * <p><blockquote>
+ * $$
+ * \ell\left(\beta, x\right) = log\left(\sum_{k=0}^{K-1} e^{margins_k -
maxMargin}\right) -
+ * margins_{y} + maxMargin
+ * $$
+ * </blockquote></p>
+ *
+ * Note that each term, $(margins_k - maxMargin)$ in the exponential is no
greater than zero; as a
+ * result, overflow will not happen with this formula.
+ *
+ * For $multiplier$, a similar trick can be applied as the following,
+ *
+ * <p><blockquote>
+ * $$
+ * multiplier_k = \left(\frac{e^{\vec{x}_i \cdot \vec{\beta}_k -
maxMargin}}{\sum_{k'=0}^{K-1}
+ * e^{\vec{x}_i \cdot \vec{\beta}_{k'} - maxMargin}} - I_{y=k}\right)
+ * $$
+ * </blockquote></p>
+ *
* @param bcCoefficients The broadcast coefficients corresponding to the
features.
* @param bcFeaturesStd The broadcast standard deviation values of the
features.
+ * @param numFeatures The number of features for the input data.
* @param numClasses the number of possible outcomes for k classes
classification problem in
* Multinomial Logistic Regression.
* @param fitIntercept Whether to fit an intercept term.
+ * @param multinomial Whether to use multinomial or binary loss
--- End diff --
multinomial (softmax) or binary loss
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