Any-Order Modeling: Methods & Applications

Updated 6 November 2025

Any-order modeling is a flexible framework that treats ordering as a latent or tunable component, enabling adaptable generation and inference across data types.
It underpins modern techniques such as AO-ARMs and masked diffusion models, which use adaptive order-policy modules and variational objectives to improve performance.
Its applications span machine learning, physics, and statistics, providing systematic methods for perturbative expansions, cumulant analysis, and order-sensitive regression.

Any-order modeling refers to a set of mathematical, algorithmic, and practical frameworks in which the generation, inference, or decomposition order is not predetermined or can be systematically varied, often treated as a latent or tunable component of the model or computation. This paradigm is increasingly central in modern machine learning (especially for generative models), physics (especially perturbative and operator formalisms), statistics (for categorical and order-sensitive data), and applied mathematics (notably in fractional calculus and numerical analysis). Any-order modeling enables tractable marginalization, flexible conditioning, robust estimation, and scalable algorithmic procedures—critical in problems where the canonical order is ambiguous, the mathematical objects are recursive, or order-agnostic inference is required.

1. Statistical, Probabilistic, and Combinatorial Foundations

Any-order modeling originates from the need to efficiently compute and represent distributions, cumulants, or regression effects over all possible orderings or subsets, rather than restricting to a single fixed path or labeling.

Autoregressive generative models provide a central example. Classic autoregressive models (ARMs) factorize $p(x) = \prod_{i=1}^L p(x_i \mid x_{<i})$ , enforcing a fixed generation order $x_1,\,x_2,\ldots,x_L$ . For data with no canonical ordering (e.g., graphs, images), this induces unnecessary bias. The "any-order" extension—Any-Order Autoregressive Models (AO-ARMs)—treats the variable order as a latent permutation $\sigma$ , uniformly or adaptively integrating over all orderings: $p_\theta(x) = \mathbb{E}_{\sigma}[ \prod_{i} p_\theta(x_{\sigma_i} | x_{\sigma_{<i}})]$ (Wang et al., 7 Mar 2025, Shih et al., 2022, Xue et al., 24 Jun 2025). This allows for flexible conditional inference and broad applicability across tasks such as masked language modeling, image inpainting, and beyond.

In combinatorics and statistics, order-invariance—or systematic order-search—arises in categorical response modeling, where the lack of a prescribed sequence among multinomial response categories motivates the model-based selection of the most predictive or theoretically supported category order, evaluated via order-averaged likelihood, AIC, or BIC (Wang et al., 2022).

2. Machine Learning: Any-Order Sequence Generation and Masked Modeling

Recent advances in language modeling, sequence modeling, and diffusion models have highlighted the power and necessity of any-order frameworks:

Fill-in LLMs (FiLM): Rather than generating tokens sequentially left-to-right or right-to-left, FiLM models (and more generally, masked diffusion models) are trained to fill in any masked position, in any order, with the context on both sides. Training employs variable masking ratios sampled from a Beta distribution, equipping the model for both infilling and unconditional generation (Shen et al., 2023). Decoding can proceed adaptively (e.g., min-entropy mask selection), with generation order conditioned on model confidence or task-specific heuristics.
Masked Diffusion Models (MDM), Any-Order GPT, and FlexMDMs: Discrete diffusion models represent sequence generation as a sequence of iterative, any-order unmasking operations, generalizing traditional decoder architectures. FlexMDMs extend this by supporting both token insertions and unmaskings, yielding parallel, any-order, and variable-length generation (Kim et al., 31 Aug 2025, Xue et al., 24 Jun 2025). The mathematical equivalence of any-order autoregressive and masked diffusion objectives underpins this unification.
Order-Policy Learning: LO-ARM models dynamically learn the generation order as a function of the current partial output, via a trainable "order-policy" module. This mechanism adapts to per-sample structure (e.g., generating graph skeletons before node attributes in molecule synthesis) and is trained with a variational lower bound (amortized ELBO), yielding significant gains in validity, uniqueness, and distributional metrics (Wang et al., 7 Mar 2025).

Comparative Table: Order Modeling in Modern ML Architectures

Modeling Paradigm	Order Adaptivity	Key Mechanism / Structure
AR/CLM	Fixed	Causal mask, left-to-right/token prefix
AO-ARM/MDM	Any-order	Masked prediction, random or learned permutations, ELBO training
LO-ARM/FiLM	Learned/Adaptive	Order-policy network, context-based ordering, variational ELBO
FlexMDM	Any-order, Variable Length	Joint insertion/unmasking, stochastic interpolant

3. Physics and Applied Mathematics: Any-Order Expansion and Systematics

High-Energy Theory (QCD, Perturbation Theory):

The derivation of fixed-order analytic formulas for jet observables under $k_t$ clustering to any perturbative order represents a breakthrough in the systematization of non-global, clustering, and phase-space logarithms. The recursive, all-orders expression involves sums and products over real/virtual emission configurations, clustering patterns, and color flows, implemented via measurement operator techniques and heavy use of step (theta) function constraints. This enables, for the first time, analytic modeling of multi-emission QCD radiation with full color structure and explicit clustering at any fixed order (Khelifa-Kerfa, 21 Sep 2024).
Similar systematic, any-order formalisms arise in cumulant analysis for heavy-ion collision experiments, where recursive algorithms (based on set partitions and Möbius inversion) generate cumulants of arbitrary order for multiparticle correlation observables—crucial for precision studies of flow, fluctuations, and detector effects (Francesco et al., 2016).

Perturbative Expansion in Dimensional Regularization:

In multi-loop Feynman integrals (“banana” diagrams), new operator-theoretic machinery, exploiting Calabi-Yau geometry and self-dual differential operators, produces ε-factorized differential systems for any loop $l$ and any expansion order in the regularization parameter ε. The algorithm constructs a canonical (ε-factorized) basis recursively, using the structure series (Y-invariants), producing all expansion terms as iterated integrals (Pögel et al., 2022).

Partial Differential Equations:

Any-order approximation theorems for solutions to Caputo-type time-fractional equations establish that, for any (possibly non-integer) order, the solution set is locally dense in smooth functions. This generalizes to ψ-Caputo operators and lays the foundation for versatile, any-order modeling in time-fractional, space-fractional, and mixed/anisotropic PDEs—essential in anomalous diffusion, viscoelasticity, and complex materials (Carbotti et al., 2018).

4. Statistical and Regression Models: Order as a Model Attribute

In generalized linear and multinomial logistic models for categorical response data, the absence of a true or known order among categories can be addressed by treating the order as a model parameter. The selection of optimal order (by AIC/BIC) is not only consistent in the presence of a true underlying order but yields substantial predictive and interpretative improvement even when no intrinsic ordering exists. This philosophy supports the broader any-order modeling principle: statistical procedures must systematically examine all possible orderings, exploiting order equivalence when present, and select or average over orders to optimize fit and inference (Wang et al., 2022).

In regression for order-of-addition experiments, flexible and parsimonious modeling strategies (e.g., response surface models based on component positions) can outperform factor-based models that encode all pairwise or position-specific order effects, and model-averaging or compound optimality design can mitigate model uncertainty (Piepho et al., 2021).

5. Algorithmic Optimization and Online Problems: Any-Order Input

Online combinatorial optimization problems, such as interval scheduling, traditionally assume a predetermined, often favorable, input order (e.g., earliest start times). Any-order input generalizes this by allowing adversarial sequence presentation. In Any-Order Online Interval Selection (AOIS), algorithms must support revocable decisions and maintain feasible solutions under worst-case orderings. Theoretical analysis yields tight competitive ratios in terms of structural instance parameters (e.g., number of distinct interval lengths), demonstrates lower-bound impossibility for memoryless/randomized algorithms beyond certain thresholds, and connects to fundamental problems in call control (Borodin et al., 2023).

6. Theoretical Implications and Future Directions

Depth/Width Tradeoffs in Representation:

Recent theory proves that two-layer, single-head transformers can represent conditional $k$ -gram models for any $k$ , solving the conditional modeling problem for arbitrarily high-order Markov chains via explicit attention and MLP constructions. This bridges gaps identified in earlier work, offering precise characterizations of the interaction between order (e.g., Markov order), model depth, and representational power (Ekbote et al., 10 Aug 2025).

Safety, Alignment, and Secure Any-Order Generation:

The flexibility of any-order generation in diffusion LLMs enlarges the attack surface for adversarial inputs. Token-level alignment strategies leveraging randomized masking and systematic supervision allow safety mechanisms (e.g., [EOS] refusals) to be enforced at any possible step, neutralizing attacks that exploit non-sequential decoding (Jeung et al., 27 Sep 2025).

7. Summary and Impact

Any-order modeling encompasses mathematical formalisms and algorithmic innovations that abandon or dynamically optimize the traditional reliance on a fixed ordering in generative, inferential, and physical computation. Whether enabling expressive and efficient generative models, systematic higher-order expansion in perturbative physics, or robust statistical modeling in ambiguous categorical settings, the paradigm provides both theoretical generality and practical versatility. It is supported by recursive or variational training objectives, explicit algorithmic generation procedures, and—where possible—order-lowering logical, algebraic, or differential operator structuring. Future research will likely continue to exploit order-adaptivity for greater modeling flexibility, robustness to uncertainty, and computational scalability across scientific and engineering disciplines.

Reference Table: Any-Order Modeling in Major Domains

Domain	Any-Order Mechanism	Key References
Generative ML (Graphs, Seqs)	AO-ARM, LO-ARM, FiLM, FlexMDM	(Wang et al., 7 Mar 2025, Shen et al., 2023, Kim et al., 31 Aug 2025)
Physics (QCD/Jets)	Recursive fixed-order formulae	(Khelifa-Kerfa, 21 Sep 2024)
High-Order Cumulants	Recursive combinatorial algorithms	(Francesco et al., 2016)
Multiloop Feynman Integrals	ε-factorized diff. systems	(Pögel et al., 2022)
Fractional PDEs	Local density for any order	(Carbotti et al., 2018)
Categorical/Ordinal Stats	Likelihood-based order selection	(Wang et al., 2022)
Online Algorithms	Adversarial input order analysis	(Borodin et al., 2023)

This broad, algorithmically tractable, and mathematically rigorous paradigm underpins much of the recent progress in domains ranging from AI to theoretical physics, and continues to catalyze new developments in both the foundations and applications of modeling, inference, and computation.