Cross-entropy Loss
Cross-entropy loss is an objective function minimized in the process of logistic regression training when a dependent variable takes more than two values.
Details
Given 
of 
,
where 
describes the class, to which the feature vector 
belongs,
where 
,
it is a matrix of size 
. The cross entropy loss objective function 
has the following format 
where
n
feature vectors
of n
p
-dimensional feature vectors, a vector of class labels ,
where describes the class, to which the feature vector belongs,
where T
is the number of classes, optimization solver optimizes cross-entropy loss objective function by argument ,
it is a matrix of size . The cross entropy loss objective function
has the following format where
,
with 
and 
, 
, 
For a given set of indices 
, 
, 
,
the value and the gradient of the sum of functions in the argument X respectively have the format:
, , ,
the value and the gradient of the sum of functions in the argument X respectively have the format:
where
Hessian matrix is a symmetric matrix of size 
, where 
, where 
, where 
is the learning rate
For more details, see [Hastie2009].
Computation
Algorithm Input
The cross entropy loss algorithm accepts the input described below.
Pass the
Input ID
as a parameter to the methods that provide input for your algorithm.
For more details, see Algorithms.Input ID | Input |
argument | A numeric table of size  with the input argument of the objective function.The sizes of the argument, gradient, and hessian numeric tables do not depend on interceptFlag .
When interceptFlag is set to false , the computation of  value is skipped,
but the sizes of the tables should remain the same. |
data | A numeric table of size  with the data  .This parameter can be an object of any class derived from NumericTable . |
dependentVariables | A numeric table of size  with dependent variables  .This parameter can be an object of any class derived from NumericTable ,
except for PackedTriangularMatrix , PackedSymmetricMatrix , and CSRNumericTable . |
Algorithm Parameters
The cross entropy loss algorithm has the following parameters.
Some of them are required only for specific values of the computation method’s parameter
method
:Parameter | Default value | Description |
algorithmFPType | float | The floating-point type that the algorithm uses for intermediate computations. Can be float or double . |
method | defaultDense | Performance-oriented computation method. |
numberOfTerms | Not applicable | The number of terms in the objective function. |
batchIndices | Not applicable | The numeric table of size  , where m is the batch size,
with a batch of indices to be used to compute the function results.
If no indices are provided, the implementation uses all the terms in the computation.This parameter can be an object of any class derived from NumericTable
except PackedTriangularMatrix and PackedSymmetricMatrix . |
resultsToCompute | gradient | The 64-bit integer flag that specifies which characteristics of the objective function to compute. Provide one of the following values to request a single characteristic or use bitwise OR
to request a combination of the characteristics:
|
interceptFlag | true | A flag that indicates a need to compute  . |
penaltyL1 | 0 | L1 regularization coefficient |
penaltyL2 | 0 | L2 regularization coefficient |
nClasses | Not applicable | The number of classes (different values of dependent variable) |
Algorithm Output
For the output of the cross entropy loss algorithm, see Output for objective functions.