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.

 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:

 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: valueValue of the objective function nonSmoothTermValueValue of non-smooth term of the objective function gradientGradient of the smooth term of the objective function hessianHessian of smooth term of the objective function proximalProjectionProjection of proximal operator for non-smooth term of the objective function lipschitzConstantLipschitz constant of the smooth term of the objective function gradientOverCertainFeatureCertain component of gradient vector hessianOverCertainFeatureCertain component of hessian diagonal proximalProjectionOfCertainFeatureCertain 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.