public class LogisticAggregator extends Object implements scala.Serializable, Logging
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 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:
$$ P(y_i=0|\vec{x}_i, \beta) = \frac{e^{\vec{x}_i^T \vec{\beta}_0}}{\sum_{k=0}^{K-1} e^{\vec{x}_i^T \vec{\beta}_k}} \\ P(y_i=1|\vec{x}_i, \beta) = \frac{e^{\vec{x}_i^T \vec{\beta}_1}}{\sum_{k=0}^{K-1} e^{\vec{x}_i^T \vec{\beta}_k}}\\ P(y_i=K-1|\vec{x}_i, \beta) = \frac{e^{\vec{x}_i^T \vec{\beta}_{K-1}}\,}{\sum_{k=0}^{K-1} e^{\vec{x}_i^T \vec{\beta}_k}} $$
The model coefficients $\beta = (\beta_0, \beta_1, \beta_2, ..., \beta_{K-1})$ 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.
$$ \begin{align} \frac{e^{\vec{x}_i^T \left(\vec{\beta}_0 + \vec{c}\right)}}{\sum_{k=0}^{K-1} e^{\vec{x}_i^T \left(\vec{\beta}_k + \vec{c}\right)}} = \frac{e^{\vec{x}_i^T \vec{\beta}_0}e^{\vec{x}_i^T \vec{c}}\,}{e^{\vec{x}_i^T \vec{c}} \sum_{k=0}^{K-1} e^{\vec{x}_i^T \vec{\beta}_k}} = \frac{e^{\vec{x}_i^T \vec{\beta}_0}}{\sum_{k=0}^{K-1} e^{\vec{x}_i^T \vec{\beta}_k}} \end{align} $$
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
$$ \begin{align} \ell\left(\beta, x_i\right) &= -log{P\left(y_i \middle| \vec{x}_i, \beta\right)} \\ &= log\left(\sum_{k=0}^{K-1}e^{\vec{x}_i^T \vec{\beta}_k}\right) - \vec{x}_i^T \vec{\beta}_y\\ &= log\left(\sum_{k=0}^{K-1} e^{margins_k}\right) - margins_y \end{align} $$
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
$$ \begin{align} \frac{\partial \ell(\beta, \vec{x}_i, w_i)}{\partial \beta_{j, k}} &= x_{i,j} \cdot w_i \cdot \left(\frac{e^{\vec{x}_i \cdot \vec{\beta}_k}}{\sum_{k'=0}^{K-1} e^{\vec{x}_i \cdot \vec{\beta}_{k'}}\,} - I_{y=k}\right) \\ &= x_{i, j} \cdot w_i \cdot multiplier_k \end{align} $$
where $w_i$ is the sample weight, $I_{y=k}$ is an indicator function
$$ I_{y=k} = \begin{cases} 1 & y = k \\ 0 & else \end{cases} $$
and
$$ multiplier_k = \left(\frac{e^{\vec{x}_i \cdot \vec{\beta}_k}}{\sum_{k=0}^{K-1} e^{\vec{x}_i \cdot \vec{\beta}_k}} - I_{y=k}\right) $$
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.
$$ \ell\left(\beta, x\right) = log\left(\sum_{k=0}^{K-1} e^{margins_k - maxMargin}\right) - margins_{y} + maxMargin $$
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,
$$ multiplier_k = \left(\frac{e^{\vec{x}_i \cdot \vec{\beta}_k - maxMargin}}{\sum_{k'=0}^{K-1} e^{\vec{x}_i \cdot \vec{\beta}_{k'} - maxMargin}} - I_{y=k}\right) $$
param: bcCoefficients The broadcast coefficients corresponding to the features. param: bcFeaturesStd The broadcast standard deviation values of the features. 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 (softmax) or binary loss
| Constructor and Description |
|---|
LogisticAggregator(Broadcast<Vector> bcCoefficients,
Broadcast<double[]> bcFeaturesStd,
int numClasses,
boolean fitIntercept,
boolean multinomial) |
| Modifier and Type | Method and Description |
|---|---|
LogisticAggregator |
add(org.apache.spark.ml.feature.Instance instance)
Add a new training instance to this LogisticAggregator, and update the loss and gradient
of the objective function.
|
Matrix |
gradient() |
double |
loss() |
LogisticAggregator |
merge(LogisticAggregator other)
Merge another LogisticAggregator, and update the loss and gradient
of the objective function.
|
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, waitinitializeLogging, initializeLogIfNecessary, isTraceEnabled, log_, log, logDebug, logDebug, logError, logError, logInfo, logInfo, logName, logTrace, logTrace, logWarning, logWarningpublic LogisticAggregator add(org.apache.spark.ml.feature.Instance instance)
instance - The instance of data point to be added.public LogisticAggregator merge(LogisticAggregator other)
this object will be modified.)
other - The other LogisticAggregator to be merged.public double loss()
public Matrix gradient()