Category-Randomized Setting

Updated 29 December 2025

Category-randomized setting is a probabilistic framework that partitions elements into distinct categories to structure stochastic selection and ensure computational tractability.
Its methodological instantiations span from stochastic choice models to adversarial-resilient randomized algorithms in distributed numerical linear algebra and optimization.
The two-stage process of category selection and within-category randomization enables robust error control and reduces sample complexity in uncertain environments.

A category-randomized setting is a probabilistic framework in which randomization or stochastic selection is structured according to a partition of the problem’s elements into distinct categories. This concept arises across a variety of mathematical and algorithmic domains, including computational mathematics, stochastic choice theory, randomized optimization algorithms, and adversarial or distributed numerical linear algebra. In all cases, category structure and randomization interact to facilitate tractable computation, statistical regularity, or robust error control under uncertainty.

1. Foundational Formalisms

Two main abstractions for category-randomized settings have been introduced in recent arXiv literature.

Stochastic choice with categorization (SCC): Here, the universe of alternatives $X$ $X$ is partitioned into $k$ $k$ disjoint categories $\mathcal{C} = \{C_1,\dots,C_k\}$ $C = {C_{1}, \dots, C_{k}}$ . A stochastic rule $p(a|A)$ $p (a ∣ A)$ over menus $A\subset X$ $A \subset X$ is said to be “category-randomized” if there exist:
- A probability vector $\pi(A)\in[0,1]^k$ selecting a category $j$ from $T(A)=\{j:C_j\cap A\neq\emptyset\}$ .
- For each $j$ , a conditional stochastic rule $\mu_j(\cdot\,|\,A\cap C_j)$ distributing over choices within $C_j$ .
- The full rule decomposes as
$p(a|A) = \sum_{j: a\in C_j} \pi_j(A) \mu_j(a | A\cap C_j).$

This structure underpins the entire class of categorization-randomized models as in "Categorize and randomize: a permissive model of stochastic choice" (Sudano, 4 Dec 2024).
Category-driven random algorithm design: In algorithmic numerical context, randomization is applied to indexed or partitioned components, e.g., selection of rows, features, or blocks, where random sampling is layered over a categorical or block partition. A notable example is adversarial-resilient randomized Kaczmarz, where redundant storage is organized by row-categories and adversarial behavior is modeled by their fractions within each group (Huang et al., 2023).

2. Core Axiomatic and Probabilistic Properties

A categorization-randomized structure is formally characterized by axioms reflecting statistical independence and neutrality between categories:

C-Independence: Choice ratios within a fixed category $C$ are invariant to the addition of outside elements, reflecting that the comparison between $a, b \in C$ is unaffected by alternatives in $X\setminus C$ .
C-Neutrality: The probability of choosing an element $x\notin C$ among $S\cup\{x\}$ depends only on alternatives outside $C$ .
Decomposability: There exists a unique maximal partition $\mathcal{C}$ rendering the process as a two-stage randomization: category then within-category choice (Sudano, 4 Dec 2024).

In distributed randomized algorithmics, the statistical quantification of error and selection in category-randomized settings uses explicit combinatorial models (e.g., mode statistics over worker categories) to bound the probability of adversarial outcomes and drive robust convergence (Huang et al., 2023).

3. Methodological Instantiations

Stochastic Choice Models

The SCC/SCWC formalism encompasses and generalizes classic discrete-choice models:

Luce model and Nested Logit: Both arise as special cases where $\pi_j(A)$ and $\mu_j$ reduce to multinomial logit forms.
Contextual and aspect-oriented extensions: Permitting $\pi_j$ and $\mu_j$ to be menu-dependent introduces flexibility to capture context effects, choice overload, and other behavioral phenomena.

Numerical and Optimization Algorithms

Adversary-tolerant randomized methods: Random sampling within row categories, combined with mode-based residual selection and statistical block-listing, ensures with high probability that updates rely on uncorrupted (majority) information, even as adversarial fractions increase.
Randomized sketching and block algorithms: Random subspace or block selection, possibly aligned with domain or variable categories, preserves statistical efficiency and algorithmic invariance in sketch-based Newton or Kaczmarz variants (Gower et al., 2019, Yesypenko et al., 2023).

4. Main Theoretical Guarantees and Error Bounds

In the stochastic choice context, category-randomization yields a series of identification and rationalizability results:

Unique coarsest partition recovery: If any nontrivial category satisfies the axioms, the SCC decomposition is unique.
RUM Rationalizability: If overall choice is RUM and SCC, the associated category and within-category rules are themselves RUMs on their respective partitions (Sudano, 4 Dec 2024).

For randomized algorithms:

Probabilistic contraction and noise floor: Mode-based selection and redundant sampling ensure contraction to arbitrarily small error depending on the fraction of adversarial categories (and zero if the noisy categories can be fully blocked), quantifiable by explicit combinatorial and spectral measures (Huang et al., 2023).
Sample complexity and error control: Randomization over categories reduces overall sample or communication complexity by leveraging redundancy and aggregation, provided categories exhibit sufficient “majority” signal.

5. Connections and Generalizations

The category-randomized setting provides a unifying abstraction for both behavioral and algorithmic randomization mechanisms:

Weak Categorization and Nonnested Partitions: Relaxing the independence of $\pi_j$ from the menu structure recovers a broader class containing standard random utility, multinomial logit, and their hierarchical or aspect-based variants.
Population and Aggregative Interpretations: Probabilistic aggregation over deterministic (resolvable) policies yields the same decompositions as direct category randomization (Sudano, 4 Dec 2024).
Distributed and Adversarial Models: The analysis and design of robust randomized algorithms naturally splits into “informative” versus “corrupted” category-driven worker classes, generalizing classical majority-vote or error-correcting schemes in the distributed learning literature (Huang et al., 2023).

6. Applications and Empirical Demonstrations

Economics and Psychology: Category-randomization models typify macro–micro decision decompositions (e.g., macro-category selection in consumer purchases) and explain empirical phenomena such as menu-effects and context-dependence (Sudano, 4 Dec 2024).
Robust Distributed Linear Algebra: Empirical studies confirm that with appropriate redundancy and block-listing, randomized Kaczmarz with category-based mode selection exhibits resilience against adversarial corruption rates exceeding 50% (Huang et al., 2023).
Statistical Machine Learning: Randomized subspace and sketching methods, when aligned with categorical domain structure, enable scalable Newton-type optimization and matrix compression with optimal sample and computational guarantees (Gower et al., 2019, Yesypenko et al., 2023).

References:

"Categorize and randomize: a permissive model of stochastic choice" (Sudano, 4 Dec 2024)
"Randomized Kaczmarz in Adversarial Distributed Setting" (Huang et al., 2023)
"RSN: Randomized Subspace Newton" (Gower et al., 2019)
"Randomized Strong Recursive Skeletonization" (Yesypenko et al., 2023)