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 \(n\) feature vectors \(X = \{x_1 = (x_{11}, \ldots, x_{1p}),\ldots, x_n = (x_{n1}, \ldots, x_{np}) \}\) of \(n\) \(p\)-dimensional feature vectors, a vector of class labels \(y = (y_1, \ldots, y_n)\), where \(y_i \in \{0, T-1\}\) describes the class, to which the feature vector \(x_i\) belongs, where \(T\) is the number of classes, optimization solver optimizes cross-entropy loss objective function by argument \(\theta\), it is a matrix of size \(T \times (p + 1)\). The cross entropy loss objective function \(K(\theta, X, y)\) has the following format \(K(\theta, X, y) = F(\theta) + M(\theta)\) where

• \(F(\theta) = -\frac{1}{n} \sum_{i=1}^{n} \log p_{y_i} (x_i, \theta) + \lambda_2 \sum_{t=0}^{T-1} \sum_{j=1}^{p} \theta_{tj}^2\), with \(p_t(z, \theta) = \frac{e^{f_t (z, \theta)}}{\sum_{i=0}^{K-1} e^{f_i (z, \theta)}}\) and \(f_t (z, \theta) = \theta_{t0} + \sum_{j=1}^{p} \theta_{tj} z_j\), \(\lambda_1 \geq 0\), \(\lambda_2 \geq 0\)

• \(M(\theta) = \lambda_1 \sum_{t=0}^{T-1} \sum_{j=1}^{p} |\theta_{tj}|\)

For a given set of indices \(I = \{i_1, i_2, \ldots, i_m \}\), \(1 \leq i_r \leq n\), \(r \in \{1, \ldots, m \}\), the value and the gradient of the sum of functions in the argument X respectively have the format:

\[F_I (\theta, X, y) = -\frac{1}{m} \sum_{i \in I} (\log p_{y_i} (x_i, \theta) + \lambda_2 \sum_{t=0}^{T-1} \sum_{j=1}^{p} \theta_{ij}^2)\]

\[\nabla F_I(\theta, x, y) = \left( \frac{\partial F_I}{\partial \theta_{00}}, \ldots, \frac{\partial F_I}{\partial \theta_{{T-1}p}} \right)^T\]

where

\[ \begin{align}\begin{aligned}\begin{split}\frac{\partial F_I}{\partial \theta_{tj}} = \begin{cases} \frac{1}{m} \sum_{i \in I} g_t (\theta, x_i, y_i) + L_{tj}(\theta), & j = 0 \\ \frac{1}{m} \sum_{i \in I} g_t (\theta, x_i, y_i) x_{ij} + L_{tj}(\theta), & j = 0 \end{cases}\end{split}\\\begin{split}g_t (\theta, x, y) = \begin{cases} p_k (x, \theta) - 1, & y = t \\ p_t (x, \theta), & y \neq t \end{cases}\end{split}\\L_{tj} (\theta) = 2 \lambda_2 \theta_{tj}\\t \in [0, T - 1]\\j \in [0, p]\end{aligned}\end{align} \]

Hessian matrix is a symmetric matrix of size \(S \times S\), where \(S = T \times (p + 1)\)

\[\begin{split}\left[\begin{array}{ccc} \frac {\partial^2 F_I} {\partial \theta_{00} \partial \theta_{00}} & \cdots & \frac {\partial^2 F_I} {\partial \theta_{00} \partial \theta_{{T-1} p}} \\ \vdots & \ddots & \vdots \\ \frac {\partial^2 F_I} {\partial \theta_{{T-1} p} \partial \theta_{00}} & \cdots & \frac {\partial^2 F_I} {\partial \theta_{{T-1} p} \partial \theta_{{T-1} p}} \end{array}\right]\end{split}\]

\[ \begin{align}\begin{aligned}\begin{split}\frac {\partial^2 F_I} {\partial \theta_{tj} \partial \theta_{pq}} = \begin{cases} \frac{1}{m} \sum_{i \in I} g_{tp} (\theta, x_i, y_i) + 2 \lambda_2, & j = 0, q = 0\\ \frac{1}{m} \sum_{i \in I} g_{tp} (\theta, x_i, y_i) x_{ij}, & j > 0, q = 0\\ \frac{1}{m} \sum_{i \in I} g_{tp} (\theta, x_i, y_i) x_{iq}, & j = 0, q > 0\\ \frac{1}{m} \sum_{i \in I} g_{tp} (\theta, x_i, y_i) x_{ij} x_{iq}, & j > 0, q > 0, j \neq q\\ \frac{1}{m} \sum_{i \in I} g_{tp} (\theta, x_i, y_i) x_{ij} x_{iq} + 2 \lambda_2, & j > 0, q > 0, j = q\\\ \end{cases}\end{split}\\\begin{split}g_{tp} (\theta, x, y) = \begin{cases} p_p (x, \theta) (1 - p_t (x, \theta)), & p = t \\ -p_t (x, \theta) p_p (x, \theta), & p \neq t \end{cases}\end{split}\\t, p \in [0, T-1]\\j, q \in [0, p]\end{aligned}\end{align} \]

\(\mathrm{prox}_\gamma^M (\theta_j) = \begin{cases} \theta_J - \lambda_1 \gamma, & \theta_j > \lambda_1 \gamma\\ 0, & |\theta_j| \leq \lambda_1 \gamma\\ \theta_j + \lambda_1 \gamma, & \theta_j < - \lambda_1 \gamma \end{cases}\), where \(\gamma\) is the learning rate

\(lipschitzConstant = \underset{i = 1, \ldots, n} \max \| x_i \|_2 + \frac{\lambda_2}{n}\)

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.

Algorithm Input for Cross-entropy Loss Computation

Input ID	Input
argument	A numeric table of size \((p + 1) \times \mathrm{nClasses}\) with the input argument \(\theta\) of the objective function. Note The sizes of the argument, gradient, and hessian numeric tables do not depend on interceptFlag. When interceptFlag is set to false, the computation of \(\theta_0\) value is skipped, but the sizes of the tables should remain the same.
data	A numeric table of size \(n \times p\) with the data \(x_ij\). Note This parameter can be an object of any class derived from NumericTable.
dependentVariables	A numeric table of size \(n \times 1\) with dependent variables \(y_i\). Note 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:

Algorithm Parameters for Cross-entropy Loss Computation

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 \(1 \times m\), 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. Note 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: value Value of the objective function nonSmoothTermValue Value of non-smooth term of the objective function gradient Gradient of the smooth term of the objective function hessian Hessian of smooth term of the objective function proximalProjection Projection of proximal operator for non-smooth term of the objective function lipschitzConstant Lipschitz constant of the smooth term of the objective function gradientOverCertainFeature Certain component of gradient vector hessianOverCertainFeature Certain component of hessian diagonal proximalProjectionOfCertainFeature Certain component of proximal projection
interceptFlag	true	A flag that indicates a need to compute \(\theta_{0j}\).
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.

Examples

C++ (CPU)Python*

• lbfgs_cr_entr_loss_dense_batch.cpp