Affine Gateaux Differentials and the von Mises Statistical Calculus

Abstract: This paper presents a general study of one-dimensional differentiability for functionals defined on convex domains that are not necessarily open. The local approximation is carried out using affine functionals, as opposed to linear functionals typically employed in standard Gateaux differentiability. This affine notion of differentiability naturally arises in certain applications and has been utilized by some authors in the statistics literature. We aim to offer a unified and comprehensive perspective on this concept.

• The paper's main contribution is establishing a rigorous framework for affine Gateaux differentiability on convex domains, enabling new mean value and envelope theorems.
• It provides detailed criteria distinguishing weak and strong affine differentiability, with explicit connections to classical Gateaux derivatives in infinite-dimensional spaces.
• The framework has practical implications in robust statistics, risk theory, optimization, and economics by clarifying sensitivity and convexity properties.

Affine Gateaux Differentials and the von Mises Statistical Calculus

Overview

The work "Affine Gateaux Differentials and the von Mises Statistical Calculus" (2403.07827) provides a comprehensive and rigorous framework for the differentiability of functionals on convex sets, particularly in settings where the domain lacks interior points—most notably spaces of probability measures. The primary innovation is the systematic study and characterization of affine Gateaux differentials, which generalize classical Gateaux differentiability by employing affine (as opposed to merely linear) local approximations. This approach yields a coherent calculus that is immediately applicable in robust statistics, risk theory, economic utility, and optimization, particularly through generalizations of mean value and envelope theorems relevant for variational analysis.

Affine Differentiability: Definitions and Main Results

The central theme of the paper is to distinguish between weak and strong affine differentiability of functionals $f : C \to \mathbb{R}$, with $C$ a convex set in a vector space $X$:

• Weak Affine Differentiability (wa-differentiability): At $x \in C$, the directional derivative $Df(x; y)$, defined via the limit along convex combinations, is affine as a function of $y \in C$.
• Affine Differentiability (a-differentiability): The weak affine differential extends to an affine functional on $X$, not only on $C$.

The paper provides sufficient criteria for wa- and a-differentiability, explores the non-uniqueness of affine extensions in infinite-dimensional and non-open domains, and relates these notions to the Gateaux derivative, proving their equivalence on the algebraic interior.

A key result (Lemma 17 and Theorem 18) establishes that wa-differentiable functionals are hemidifferentiable along line segments—this yields a mean value theorem in the affine framework, formally analogous to the standard mean value theorem, but tailored to spaces without interior.

Calculus, Convexity, and Subdifferential Structure

The paper constructs an affine calculus that includes product, chain, and sum rules for wa-differentiable maps, as well as the interaction with convexity:

• Convexity Characterization: The monotonicity of the affine differential characterizes convexity of the functional, and the paper provides necessary and sufficient conditions for convexity in terms of $Df(x; y)$.
• Subdifferential Relationship: For convex, a-differentiable functionals, the subdifferential at $x$ is explicitly characterized in terms of the affine gradient and the normal cone: $\partial f(x) = x^* + N_C(x)$. At internal points, the subdifferential reduces exactly to the affine gradient.

The extension to strict, Hadamard, and Fréchet versions of affine differentiability is systematically presented, leveraging generalized metrics (not limited to norms), further supporting robust analysis in infinite-dimensional and non-linear functional spaces.

Applications in Statistics, Optimization, and Economics

The methodology finds direct application in several key domains:

• Influence Functions and Robust Statistics: The von Mises influence function is naturally interpreted as an affine differential. The paper clarifies when an influence function corresponds to an affine (not just weak affine) differential, resolving known pathologies (Examples 32–33).
• Envelope Theorems and Optimization: A Danskin-type theorem (Theorem 31) is established for wa-differentiable functionals, delivering sharp sensitivity results for optimization problems where the objective is the pointwise supremum over a family of parameterized functionals.
• Risk Theory and Utility: The calculus is used to localize expected utility functionals for quadratic and prospect theory examples, identifying precise gradient structures for mixture-based utility models. The approach unifies the treatment of differentiability in classical, rank-dependent, and cumulative prospect theory settings.
• Bayesian Robustness: Affine differentials facilitate robust Bayesian analysis by providing necessary conditions for the sensitivity of posterior statistics to prior perturbations. The calculus supports the derivation of explicit formulas for the derivative of functionals such as the posterior mean and expected posterior loss with respect to the prior, even for nonparametric models.

Generalizations: Strict, Hadamard, and Fréchet Affine Differentiability

The authors extend the calculus by introducing Hadamard and Fréchet variants of affine differentiability, underpinned by generalized convex and homogeneous metrics (including Prokhorov and Dudley metrics on probability spaces).

• The framework proves that boundedly wa-differentiable functionals are Hadamard wa-differentiable under convex metrics, and strictly Fréchet wa-differentiable on the interior, analogously to the classical setting.
• Detailed examples in both finite and infinite-dimensional spaces (e.g., quadratic forms on probability measures) illustrate these concepts, highlighting the subtleties in the extension to non-open, infinite-dimensional convex domains.

Implications and Directions

The framework has immediate implications for variational analysis, statistical asymptotics, robust estimation, economic choice under ambiguity, and stochastic optimization:

• Statistical Asymptotics: By ensuring that differentiable structure extends to convex sets of probability distributions, the approach guarantees the regularity required for CLT-style results for plug-in and $M$-estimators under weak assumptions.
• Optimization and Sensitivity: The robustness of optimization solutions to perturbations of priors, loss functions, or constraints can be analyzed with high precision, even in settings with non-classical domains or lack of linearity.
• Modeling Economics under Ambiguity: The multi-utility and local expected utility representations derived from affine differentials support the design and analysis of decision models under ambiguity and imprecision, connecting with Machina-style local expected utilities and related robust preference frameworks.

Future developments may include extending the affine calculus to more general set-valued functionals, stochastic processes indexed by general state spaces, and integrating these derivatives into computational algorithms for robust optimization and inference.

Conclusion

This work advances the functional analytic foundation for differentiability on convex (and possibly boundary-less) domains, systematizing and unifying the scattered concept of affine (von Mises) differentials with a rigorous calculus and broad applicability. By aligning well with practical needs in statistics, economics, and optimization, this theory is poised to underpin further methodological and computational advances in fields requiring robust, non-linear functional sensitivity analysis (2403.07827).