Sum of Functions

The sum of functions \(F(\theta)\) is a function that has the form of a sum:

\[F(\theta) = \sum _{i=1}{n} F_i(\theta), \theta \in \mathbb{R}^p\]

For given set of the indices \(I = \{i_1, i_2, \ldots , i_m\}\), \(1 \leq ik < n\), \(k \in \{1, \ldots, m\}\), the value and the gradient of the sum of functions in the argument \(\theta\) has the format:

\[F_I(\theta) = \sum _{i \in I} F_i(\theta)\]
\[\nabla_I F_I(\theta) = \sum _{i \in I} \nabla F_i(\theta)\]

The set of the indices \(I\) is called a batch of indices.

Computation

Algorithm Input

The sum of functions 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 \(p \times 1\) with the input argument of the objective function.

Algorithm Parameters

The sum of functions algorithm has the following parameters:

Parameter

Default Value

Description

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

Algorithm Output

For the output of the sum of functions algorithm, see Output for objective functions.