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Linear Functions to the Extended Reals

Bo Waggoner

TL;DR

The paper develops a rigorous framework for linear extended functions $f:\mathbb{R}^d\to\overline{\mathbb{R}}$, showing they are exactly convex and concave with $f(\vec{0})=0$ and admitting a dimensionally recursive, $\Omega(d^2)$-parameter representation. It introduces the notions of extended subgradients and affine extended supports, proving that every convex function with a convex effective domain has extended subgradients at every finite point and that convexity is equivalent to being the supremum of affine extended functions. The work then connects this theory to proper scoring rules, providing a constructive method to build scoring rules from a given convex function and clarifying when such rules are (strictly) proper under broad conditions. Overall, the results offer a concrete, algebraic handle on extended-valued convex analysis and its applications to scoring rules, including a constructive path from convex functions to extended subgradients and vertical supporting hyperplanes to epigraphs.

Abstract

This paper investigates functions from $\mathbb{R}^d$ to $\mathbb{R} \cup \{\pm \infty\}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and unlike finite linear functions, they require $Ω(d^2)$ parameters to uniquely identify. In particular they can capture vertical tangent planes to epigraphs: a function (never $-\infty$) is convex if and only if it has an extended-valued subgradient at every point in its effective domain, if and only if it is the supremum of a family of "affine extended" functions. These results are applied to the well-known characterization of proper scoring rules, for the finite-dimensional case: it is carefully and rigorously extended here to a more constructive form. In particular it is investigated when proper scoring rules can be constructed from a given convex function.

Linear Functions to the Extended Reals

TL;DR

The paper develops a rigorous framework for linear extended functions , showing they are exactly convex and concave with and admitting a dimensionally recursive, -parameter representation. It introduces the notions of extended subgradients and affine extended supports, proving that every convex function with a convex effective domain has extended subgradients at every finite point and that convexity is equivalent to being the supremum of affine extended functions. The work then connects this theory to proper scoring rules, providing a constructive method to build scoring rules from a given convex function and clarifying when such rules are (strictly) proper under broad conditions. Overall, the results offer a concrete, algebraic handle on extended-valued convex analysis and its applications to scoring rules, including a constructive path from convex functions to extended subgradients and vertical supporting hyperplanes to epigraphs.

Abstract

This paper investigates functions from to that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on , and unlike finite linear functions, they require parameters to uniquely identify. In particular they can capture vertical tangent planes to epigraphs: a function (never ) is convex if and only if it has an extended-valued subgradient at every point in its effective domain, if and only if it is the supremum of a family of "affine extended" functions. These results are applied to the well-known characterization of proper scoring rules, for the finite-dimensional case: it is carefully and rigorously extended here to a more constructive form. In particular it is investigated when proper scoring rules can be constructed from a given convex function.

Paper Structure

This paper contains 19 sections, 11 theorems, 8 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proper scoring rules.
  3. Preliminaries.
  4. The extended reals.
  5. Linear extended functions
  6. Proof.
  7. Proof.
  8. Proof.
  9. Proof.
  10. Proof.
  11. Structure.
  12. Extended subgradients
  13. Proof.
  14. Proof.
  15. Proof.
  16. ...and 4 more sections

Key Result

Proposition 2.3

$f: \mathbb{R}^d \to \overline{\mathbb{R}}$ is linear extended if and only if $f(\vec{0}) = 0$ and $f$ is both concave and convex.

Figures (1)

  • Figure 1: The domains of two rank-$2$ linear extended functions on $\mathbb{R}^2$. In each case, the function is defined as: $f(x) = \infty \cdot \textrm{sign}(v_1 \cdot x)$ if $v_1 \cdot x \neq 0$, otherwise $f(x) = \infty \cdot \textrm{sign}(v_2 \cdot x)$. The shaded orange region is $f^{-1}(\infty)$, while $f(\vec{0}) = 0$ and the remaining space is $f^{-1}(-\infty)$. Coordinate axes, gray, are thickened to illustrate the behavior on those subspaces. When $f$ represents a vertical tangent plane to a convex function $g$, then $\textrm{effdom}\mathopen{}\mathclose{\left(g\right)$ must lie in the non-orange region.

Theorems & Definitions (18)

  • Definition 2.1
  • Claim 2.2
  • Proposition 2.3: c.f. dudik2022convex
  • proof
  • Lemma 2.4: Decomposition
  • Definition 2.5: Rank
  • Lemma 2.6: Recursive definition
  • Proposition 2.7: Correctness of Algorithm \ref{['alg:linext']}
  • Proposition 2.8: Parsimonious parameterization
  • Definition 3.1: Extended subgradient
  • ...and 8 more