Stratified Normalization Methods

• Stratified normalization is a technique that applies localized normalization within partitioned strata defined by categorical or structural properties.
• It employs iterative and modular algorithms—such as local test pair procedures and level-indexed techniques—to optimize performance compared to global normalization methods.
• This approach underpins advancements in fields like algebraic geometry, machine learning, reinforcement learning, and logical frameworks by addressing heterogeneity and improving stability.

Stratified normalization refers to a range of mathematical, logical, statistical, and algorithmic techniques that enforce normalization processes within distinct, intrinsically defined strata—subsets partitioned by categorical, indexical, or structural properties of the underlying domain. By carrying out normalization, correction, or evaluation locally within each stratum, these methods address heterogeneity and complexity that would compromise accuracy, computational efficiency, or theoretical guarantees if treated globally. Applications occur in commutative algebra, logic, type theory, machine learning, particle filtering, reinforcement learning, and algebraic geometry. The principle is to separate (“stratify”) components so the normalization is both well-defined and optimized for the specific structure or statistical properties of each stratum.

1. Algebraic Stratified Normalization

In the context of affine algebras over a perfect field, stratified normalization is a parallelizable algorithmic framework for computing the normalization $\bar{A}$ of a reduced affine algebra $A$ (Boehm et al., 2011). The singular locus $\operatorname{Sing}(A)$, defined as $$\{P \in \operatorname{Spec}(A) \mid A_P \text{ not regular}\}$$, is partitioned into strata $V_L$ according to combinatorial properties of the minimal primes containing $P$.

Normalization is applied locally: for each stratum $V \subseteq \operatorname{Sing}(A)$, an intermediate ring $A^{(V)}$ is computed by an iterative procedure involving test pairs and the Grauert–Remmert criterion, yielding chains of extensions

$$A = A_0 \subseteq A_1 \subseteq \cdots \subseteq A_m = A^{(V)},$$

where each $A_{i+1}$ is computed as $\mathrm{Hom}_A(J_i, J_i)$ and $J_i$ is the radical of the extended ideal. The aggregate normalization is given by

$\bar{A} = \sum_{V \in \mathrm{Strata}(A)} A^{(V)}.$

This partitioning enables both modular computation (lifting results from finite field computations via Chinese Remainder Theorem and rational reconstruction) and parallelization, leading to substantial improvements in computational complexity and performance relative to global algorithms.

2. Logical and Type-Theoretic Stratification

Stratified normalization in logic refers to systems (notably in linear logic and set theory) in which formulas, terms, or rewrites are indexed by stratification levels (Boudes et al., 2012, Gabbay, 2017, Dowek, 2023, Jacob-Rao et al., 2018). In stratified linear logic, normalization (specifically cut-elimination) is regulated via a paragraph modality $\mathsf{S}$ that tracks the “level” of formulas, ensuring interactions—e.g., contraction and cut—are only performed within the same stratum. This yields strong normalization properties: for all well-formed derivations, reduction sequences terminate and normal forms are unique.

In type theory (e.g., the Tores language), index-stratified types are built by recursion on indices. For instance, the definition

$$T_{\text{Rec}}(0) = T_0,\quad T_{\text{Rec}}(\text{suc }M) = T_s[M/u,\,T_{\text{Rec}}M/X]$$

ensures all type-level recursion unfolds along a well-founded measure, guaranteeing termination and soundness (Jacob-Rao et al., 2018). These techniques enable metatheoretic normalization proofs, including normalization by evaluation for lambda calculi.

3. Stratified Normalization in Machine Learning

In statistical learning and deep networks, stratified normalization denotes normalization schemes applied to groups determined by intrinsic characteristics (e.g., batch, channel, or category) (Hoffer et al., 2018, Tuck et al., 2020, Zhu et al., 7 Oct 2025). Batch Normalization (BN) stratifies the normalization of activations per channel via $L^2$, $L^1$, or $L^\infty$ metrics, decoupling scale from direction and thereby improving numerical stability, especially for low-precision arithmetic. Formally, $L^1$ BN computes

$$\hat{x}^{(k)} = \frac{x^{(k)} - \mu_k}{C_{L_1}(1/n)\|x^{(k)} - \mu_k\|_1},$$

with $C_{L_1} = \sqrt{\pi/2}$ chosen for correct scaling under Gaussian assumptions.

Eigen-stratified models for categorical variables constitute another advanced approach: for a categorical variable with $K$ levels, parameters are constrained to lie in the span of the bottom $m$ eigenvectors of the Laplacian of a graph defined on category labels. This dramatically reduces the number of free parameters and regularizes estimates, particularly beneficial when $K$ is large and data per category is sparse. The parameterization

$\theta = Z\tilde{Q}^\top$

and Laplacian regularizer

$$\mathcal{L}(\theta) = \frac{1}{2}\|Z\Lambda_m^{1/2}\|_F^2$$

enforce smoothness and constrain the model complexity (Tuck et al., 2020).

4. Stratified Normalization in Particle Filters and Monte Carlo Methods

Stratified normalization arises in particle filtering via the stratified resampling mechanism (Flenghi et al., 2023), where normalized weights are partitioned based on the integer and fractional components of particle allocations. Analysis proves that the fractional parts of partial sums of normalized weights converge in distribution to a uniform random variable on $[0,1]$, crucial for asymptotic normality. The central limit theorem for stratified resampling yields asymptotic variance formulas that differentiate Monte Carlo variability and stratification-induced variance, facilitating error control: $$\sqrt{M} \left( \frac{1}{M} \sum_m f(Y_m^M) - \nu(fg) \right) \rightarrow \mathcal{N}(0, \sigma^2(f) + \tilde{\sigma}^2(f)),$$ with explicit decomposition of variance components.

This stratified approach enables quantification of Monte Carlo error when stratification decouples dependencies, providing practical guidelines for variance reduction and reliability in sequential inference schemes.

5. Stratified Advantage Normalization in Reinforcement Learning

In reinforcement learning for heterogeneous environments (notably, training LLM agents for search-based tasks), stratified normalization is employed via Stratified Advantage Normalization (SAN) (Zhu et al., 7 Oct 2025). Here, structurally heterogeneous trajectories (e.g., varying by number of search calls) are partitioned into homogeneous strata. SAN computes per-trajectory advantage via

$$A_{\text{SAN}}(\tau) = \frac{R(\tau) - \hat\mu_k(x)}{\hat\sigma_k(x) + \varepsilon}$$

where $\hat\mu_k(x)$ and $\hat\sigma_k(x)$ are the empirical mean and standard deviation in stratum $k$.

SAN eliminates cross-stratum bias—the distortion caused by evaluating heterogeneous trajectories with a global baseline. The paper proves that SAN yields conditionally unbiased, unit-variance estimates within each stratum and corrects for bias present in global baselines. Empirically, SAN results in more stable and effective policy optimization, outperforming classic global methods in multi-hop question answering tasks—establishing stratification as a principled remedy for structural heterogeneity in policy gradient methods.

6. Stratified Normalization in Foundations and Semantics

Deductive systems and set-theoretic foundations utilize stratified normalization to ensure strong normalization and consistency (Dowek, 2023, Gabbay, 2017). In the Stratified Foundations theory constructed via deduction modulo, normalization is guaranteed by constructing a pre-model interpreted over an $\omega$-model with level-shifting automorphisms. Rewrite rules reflecting comprehension schemes and membership are tightly controlled via stratification, ensuring that every proof reduces finitely and denotational semantics are well-defined.

This approach reinforces the utility of stratification in logical frameworks, particularly to secure foundational properties (strong normalization, confluence, uniqueness of normal forms) critical for soundness and mechanizability in automated theorem proving.

7. Applications in Algebraic Geometry: Normalization of Strata

Normalization of closed Ekedahl–Oort strata in Shimura varieties is achieved by lifting the strata to partial flag spaces and recovering normalization via Stein factorization (Koskivirta, 2017). Stratification is determined by parabolic subgroups associated with canonical filtrations of Barsotti–Tate groups. The normalization of the Zariski closure of a stratum $\overline{Z}_w$ is given by

$$\overline{Z}_w^{\text{norm}} \simeq \operatorname{Spec}(\overline{Z}_{P_w, w}),$$

where $P_w$ is the canonical parabolic reflecting the stabilizer of the canonical filtration. This explicit construction elucidates the interplay between group-theoretic stratification and normalization, resolving singularities and yielding canonical desingularizations relevant to moduli space and arithmetic geometry.

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Stratified normalization, across these diverse domains, consistently addresses complex heterogeneity and structural dependencies by partitioning and localizing the normalization procedure. Whether in algebraic computation, statistical learning, logical deduction, or reinforcement learning, stratified normalization underpins efficient algorithms and robust theoretical guarantees, serving as a central principle for overcoming limitations imposed by global or "coarse" normalization approaches.