Role-Aware Bucket Aggregation

• Role-aware bucket aggregation is a design principle that partitions objects into semantically coherent buckets based on functional, semantic, or structural roles.
• It addresses aggregation failures by separating detection from execution, ensuring consistency across diverse settings such as federated learning and mechanistic interpretability.
• Practical implementations show that role-aware methods preserve intended invariants, reduce cross-modal interference, and maintain causal significance in complex aggregations.

Searching arXiv for the cited papers to ground the article in current records. arxiv_search query="(Jiang, 30 May 2026) Detection vs. Execution: Single-Bucket Probes Miss Half the Mamba-2 State Sink" max_results=5 sort_by="relevance" sort_order="descending" arxiv_search query="(Wang et al., 26 Sep 2025) Role-Aware Multi-modal federated learning system for detecting phishing webpages" max_results=5 sort_by="relevance" sort_order="descending" arxiv_search query="(Huang et al., 2023) Aggregation Consistency Errors in Semantic Layers and How to Avoid Them" max_results=5 sort_by="relevance" sort_order="descending" arxiv_search query="(Loo, 2024) Split-Apply-Combine with Dynamic Grouping" max_results=5 sort_by="relevance" sort_order="descending" Role-aware bucket aggregation denotes aggregation procedures in which the objects being combined are first partitioned into buckets whose meaning depends on a functional, semantic, or structural role, and are then aggregated in a way that preserves those role distinctions rather than collapsing them into a single undifferentiated average. In the literature summarized here, the phrase is used explicitly for multimodal federated learning (Wang et al., 26 Sep 2025), while in mechanistic interpretability it is a precise description of a central result on Mamba-2 even though the paper does not use the phrase itself (Jiang, 30 May 2026). Related work supplies transferable formal machinery for normalization under joins, adaptive fallback under sparsity, provenance of aggregate values, and ordered bucket structures for ranking data (Huang et al., 2023, Loo, 2024, Amsterdamer et al., 2011, Achab et al., 2018, Aledo et al., 19 Feb 2025).

1. Conceptual scope and bucket semantics

Across these works, a “bucket” is not a single ontological object. It may be a probe-induced class-membership partition over model units, a join-key group in a semantic layer, a dynamically collapsed analysis group, a parameter subset associated with a modality expert, or an ordered partition of ranked items. The common thread is that aggregation is treated as correct only relative to the role played by the grouped objects.

Setting	Bucket object	Role distinction
Mechanistic interpretability (Jiang, 30 May 2026)	specialist, dual, generalist, silent unit sets	execution vs detection
Semantic layers (Huang et al., 2023)	join-key groups	base-metric vs exploratory relations
Dynamic grouping (Loo, 2024)	target groups and collapsed support groups	finest target vs fallback subset
Federated learning (Wang et al., 26 Sep 2025)	image, html, url, fusion, shared parameter groups	modality expert roles
Ranking aggregation (Achab et al., 2018, Aledo et al., 19 Feb 2025)	ordered item buckets or sets of bucket orders	within-bucket variability vs between-bucket precedence

This multiplicity matters because different papers diagnose different failure modes of naive aggregation. In semantic layers, the failure is inconsistency under join fanout. In Mamba-2 interpretability, the failure is conflating detection and execution because a single-bucket specialty rule discards multi-class units. In multimodal federated learning, the failure is “cross-modal gradient interference” and “ineffective averaging” when fundamentally different embeddings are averaged together. In ranking problems, the failure is over-compression to one consensus order when the data support several bucketed order structures (Huang et al., 2023, Jiang, 30 May 2026, Wang et al., 26 Sep 2025, Aledo et al., 19 Feb 2025).

A useful synthesis is that role-aware bucket aggregation is less a single algorithm than a design principle: define buckets so that the grouped object remains semantically coherent, then apply an aggregation operator whose invariants match the intended role semantics. The cited papers differ in implementation, but they converge on the claim that aggregation is not merely bookkeeping.

2. Probe-induced buckets in mechanistic interpretability

In the Mamba-2 study, the probed objects are $(\text{layer}, \text{head})$ units, while in Mamba-1 they are $(\text{layer}, \text{channel})$ units. The probed mechanism is the state sink, defined as disproportionately high token-conditional $\Delta$-gate activation on boundary tokens such as BOS and newline, presented as the selective-state-space analogue of the Transformer attention sink (Jiang, 30 May 2026).

The representational signature is based on mean pre-softplus $\Delta$-gate activation, denoted $\bar a_{\ell,u,c}$, over token classes

$$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$

For each unit, the paper computes a class-conditional differential $\Delta \bar a_{\ell,u,c}$ relative to a reference non-class token set, and thresholds these differentials at the fixed pre-registered threshold

$\tau = 0.5,$

chosen because $\mathrm{softplus}(0.5) \approx 0.97$, approximately the gate’s “on” state. This yields the binary specialty matrix

$S_{\ell,u,c} \in \{0,1\}.$

Units are then partitioned by counting how many classes exceed threshold. A single-bucket $(\text{layer}, \text{channel})$0-specialist has exactly one active class and it is $(\text{layer}, \text{channel})$1. A dual specialist has exactly two active classes. A generalist has three or more, and a silent unit has none. In the paper’s own terminology, a single-bucket probe retains only exactly-one-class units; multi-class aggregation recovers units with richer signatures, especially the dual heads (Jiang, 30 May 2026).

This formalism is notable because the bucket assignment is not a learned classifier over logits. It is a thresholded class-membership rule over $(\text{layer}, \text{channel})$2. The methodological consequence is that bucket construction is explicit and auditable: bucket membership is determined by the binary specialty matrix, not by a latent clustering step. A plausible implication is that the paper treats bucket aggregation as a first-order experimental object rather than a post hoc visualization.

3. Detection and execution in the Mamba-2 state sink

The central empirical finding is that the Mamba-2 state sink decomposes into two functional head sets at the same depth: single-bucket BOS-specialist heads form an execution layer, while dual heads form a detection layer (Jiang, 30 May 2026). The distinction is not small. For Mamba-2, the reported fractions are: 130M, bos-specialist $(\text{layer}, \text{channel})$3 and dual $(\text{layer}, \text{channel})$4; 1.3B, bos-specialist $(\text{layer}, \text{channel})$5 and dual $(\text{layer}, \text{channel})$6; 2.7B, bos-specialist $(\text{layer}, \text{channel})$7 and dual $(\text{layer}, \text{channel})$8. Thus, at 2.7B, the single-bucket BOS set is about $(\text{layer}, \text{channel})$9 of heads, whereas the dual set is $\Delta$0.

Representation-level statistics alone favor the dual heads. The class-conditional cosine analysis yields

$\Delta$1

for dual heads, versus

$\Delta$2

for single-bucket BOS specialists. On M-2 1.3B the values are dual $\Delta$3 and bos_spec $\Delta$4; on M-2 2.7B they are dual $\Delta$5 and bos_spec $\Delta$6. The paper also reports that a linear probe on concatenated $\Delta$7 vectors predicts the 4-way next-token boundary class nearly perfectly for both sets: dual $\Delta$8 on M-2 2.7B and $\Delta$9 on M-2 1.3B, bos-specialist $\Delta$0 on both. Representational similarity and linear decodability therefore do not separate the roles.

Causal intervention does. The main causal score is the NLL differential under ablation,

$\Delta$1

written in Table 1 as

$\Delta$2

where $\Delta$3 is the mean over 30 random size-matched complement seeds. Positive values mean the specialist bucket is more causally important than a random equal-size set. The strongest summary in the data is that BOS-specialists are positively causal on both BOS-context and newline-target predictions in all six scale/corpus cells, whereas dual heads underperform on newline in all six. At M-2 2.7B, for example, bos-spec on wt_nl is $\Delta$4 on wt-2 and $\Delta$5 on Pile, while dual on wt_nl is $\Delta$6 on wt-2 and $\Delta$7 on Pile. Appendix F1 reports $\Delta$8 CIs for negative newline dual effects, including $\Delta$9, CI $\bar a_{\ell,u,c}$0, for M-2 2.7B wt-2 (Jiang, 30 May 2026).

The behavioral consequences are correspondingly sharp. On RULER Needle-in-a-Haystack retrieval at 1024 context length, ablating the BOS-specialist set collapses retrieval accuracy from $\bar a_{\ell,u,c}$1 to $\bar a_{\ell,u,c}$2 in both Mamba-1 2.8B and Mamba-2 2.7B, while size-matched complements preserve baseline accuracy $\bar a_{\ell,u,c}$3. The per-token NLLs at 1024 are M-1 2.8B baseline $\bar a_{\ell,u,c}$4, bos-spec $\bar a_{\ell,u,c}$5, complement $\bar a_{\ell,u,c}$6; and M-2 2.7B baseline $\bar a_{\ell,u,c}$7, bos-spec $\bar a_{\ell,u,c}$8, complement $\bar a_{\ell,u,c}$9. A magnitude-matched non-specialist complement in Mamba-2 2.7B still preserves retrieval, with 1024 accuracy $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$0, NLL $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$1, and 2048 accuracy $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$2, NLL $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$3. Dual-head ablation produces only partial degradation: at 1024, accuracy $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$4, NLL $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$5; at 2048, accuracy $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$6, NLL $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$7. The paper’s conclusion is explicit: probe-derived specialty can identify execution circuits, but at coarse granularity the same probe also recovers detection circuits, and separating them requires class-conditional ablation rather than class-conditional cosine (Jiang, 30 May 2026).

A common misconception addressed directly in the paper is that the discrepancy is merely a granularity artifact. The T3 control on Mamba-1 2.8B groups channels into random buckets of width $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$8. At $$c \in \{\text{BOS}, \text{newline}, \text{punctuation}, \text{whitespace\_word}\}.$$9, where the effective unit budget per class matches Mamba-2 2.7B’s $\Delta \bar a_{\ell,u,c}$0, the random buckets still yield bos-spec around $\Delta \bar a_{\ell,u,c}$1–$\Delta \bar a_{\ell,u,c}$2 and dual around $\Delta \bar a_{\ell,u,c}$3–$\Delta \bar a_{\ell,u,c}$4, with $\Delta \bar a_{\ell,u,c}$5 specifically giving bos-spec $\Delta \bar a_{\ell,u,c}$6 and dual $\Delta \bar a_{\ell,u,c}$7, unlike real Mamba-2 2.7B with bos-spec $\Delta \bar a_{\ell,u,c}$8 and dual $\Delta \bar a_{\ell,u,c}$9. Phase B ablation on bucketed Mamba-1 gives only sub-nat differentials. The paper therefore argues that the phenomenon is not due merely to substrate granularity and is instead consistent with Mamba-2’s head-shared $\tau = 0.5,$0 projection (Jiang, 30 May 2026).

4. Consistency, dynamic grouping, and aggregate provenance

Outside interpretability, the most direct formal precursor is the semantic-layer literature on aggregation consistency. In that setting, joins alter tuple multiplicities, so naive aggregation over a joined result no longer preserves the intended metric semantics. The proposed remedy is “weighing”: assign equal total weight to each join-key group rather than each tuple, then distribute that group weight among the tuples in the group. The central sufficient condition is

$\tau = 0.5,$1

under which the enriched query preserves the base aggregate because multiplication by $\tau = 0.5,$2 is identity in the semiring. With selections, the preservation law becomes

$\tau = 0.5,$3

Equal weighing, order-based weighing, position-based weighing, and proportional weighing are all discussed, and equal weighing is implementable as $(\text{layer}, \text{channel})$30 (Huang et al., 2023)

Dynamic grouping addresses a different failure mode: fixed groups may be too sparse for reliable aggregation. The formal model keeps the target grouping fixed while allowing the data used to compute each target’s aggregate to expand adaptively along a user-defined collapse sequence

$\tau = 0.5,$4

For a target group $\tau = 0.5,$5, the candidate subset at collapse level $\tau = 0.5,$6 is

$\tau = 0.5,$7

and the algorithm chooses the smallest $\tau = 0.5,$8 such that $\tau = 0.5,$9. The output is a triple $\mathrm{softplus}(0.5) \approx 0.97$0, where the target label is preserved and the level field records how far the evidence had to collapse. The complexity is $\mathrm{softplus}(0.5) \approx 0.97$1 in the best case and $\mathrm{softplus}(0.5) \approx 0.97$2 in the worst case, and the method reduces exactly to ordinary split-apply-combine when $\mathrm{softplus}(0.5) \approx 0.97$3 (Loo, 2024).

The provenance literature supplies a more algebraic perspective. Green, Karvounarakis, and Tannen’s semiring-provenance framework is shown to be insufficient for aggregation if provenance is attached only to tuples, because aggregate values depend on the multiset of contributing values and the desired semantics should commute with homomorphisms. The proposed solution is to annotate not only tuples but also the individual values inside aggregates, using tensor expressions such as

$\mathrm{softplus}(0.5) \approx 0.97$4

For GROUP BY, the grouped tuple exists with a $\mathrm{softplus}(0.5) \approx 0.97$5-annotation, while the aggregate field records the contributor expression. This yields a formal analogue of bucket internals: not just whether a grouped output exists, but how its value was assembled from contributing records (Amsterdamer et al., 2011).

Taken together, these papers suggest three orthogonal requirements for role-aware bucket aggregation. First, bucket weights should preserve the relevant invariant, such as total metric mass under joins. Second, evidence buckets may need to expand adaptively under support constraints without changing the target bucket identity. Third, aggregate outputs may need internal provenance if the goal is to explain which contributors or subroles formed the bucketed value. None of these papers defines “role-aware” in the strong sense used later in federated learning, but they supply the normalization, fallback, and provenance primitives from which such systems can be built (Huang et al., 2023, Loo, 2024, Amsterdamer et al., 2011).

5. Ordered buckets and set-valued aggregation in ranking problems

In ranking data, bucket aggregation appears as ordered partitioning. One line of work defines a bucket order $\mathrm{softplus}(0.5) \approx 0.97$6 as an ordered partition of the item set, with deterministic precedence across buckets and unrestricted ranking distributions within each bucket. For a ranking distribution $\mathrm{softplus}(0.5) \approx 0.97$7, the canonical bucket projection preserves all within-bucket ranking variability while enforcing the bucket order across buckets. Under Kendall’s $\mathrm{softplus}(0.5) \approx 0.97$8, the distortion of a bucket order has the closed form

$\mathrm{softplus}(0.5) \approx 0.97$9

the total probability mass of inter-bucket inversions. Empirically this becomes

$S_{\ell,u,c} \in \{0,1\}.$0

Under strong and strict stochastic transitivity, the optimal fixed-shape bucket order is obtained by slicing the Kemeny median into contiguous segments, and the paper also proposes the bottom-up BuMeRank procedure that merges adjacent buckets according to the distortion reduction $S_{\ell,u,c} \in \{0,1\}.$1 (Achab et al., 2018).

A second line of work starts from the Optimal Bucket Order Problem, where the output is a single bucket order represented by a bucket matrix $S_{\ell,u,c} \in \{0,1\}.$2 with entries $S_{\ell,u,c} \in \{0,1\}.$3, $S_{\ell,u,c} \in \{0,1\}.$4, or $S_{\ell,u,c} \in \{0,1\}.$5, and the objective is

$S_{\ell,u,c} \in \{0,1\}.$6

for a pair order matrix $S_{\ell,u,c} \in \{0,1\}.$7. The generalization, OSBOP, replaces the single output with a weighted set of $S_{\ell,u,c} \in \{0,1\}.$8 bucket orders. The weighted matrix average is

$S_{\ell,u,c} \in \{0,1\}.$9

and the objective is

$(\text{layer}, \text{channel})$00

On 14 PrefLib datasets with $(\text{layer}, \text{channel})$01, the paper reports average ratio OSBOP$(\text{layer}, \text{channel})$02/OBOP $(\text{layer}, \text{channel})$03, OSBOP$(\text{layer}, \text{channel})$04/OBOP $(\text{layer}, \text{channel})$05, and OSBOP$(\text{layer}, \text{channel})$06/OSBOP$(\text{layer}, \text{channel})$07 $(\text{layer}, \text{channel})$08, corresponding to average improvements of about $(\text{layer}, \text{channel})$09, $(\text{layer}, \text{channel})$10, and $(\text{layer}, \text{channel})$11 respectively (Aledo et al., 19 Feb 2025).

These ranking formulations are important because they make explicit a distinction that also recurs in other domains: a single bucketed summary may be too coarse, while a small set of bucketed structures can preserve heterogeneity without abandoning interpretability. In the transport-based framework, within-bucket uncertainty is retained and only inter-bucket violations are penalized. In OSBOP, several bucket orders can coexist, with weights functioning as mixture masses. This suggests a role-aware interpretation in which roles are ordered strata or latent preference modes rather than rigid equivalence classes. The papers themselves do not define explicit role variables, but they provide two complementary models of bucket structure: one ordered partition with internal flexibility, and one set-valued consensus over several bucket orders (Achab et al., 2018, Aledo et al., 19 Feb 2025).

6. Multimodal federated learning as an explicit role-aware implementation

The phishing-detection paper uses the term “role-aware bucket aggregation” directly for a server-side strategy layered on top of FedProx. The model contains four expert routes—IMAGE head, HTML head, URL head, and Fusion head—and the server partitions parameters into role buckets

$(\text{layer}, \text{channel})$12

Hard gating replaces learnable routing: the modality label directly selects the expert during local training, and the same role label determines which clients are eligible to contribute to each parameter during aggregation (Wang et al., 26 Sep 2025).

The local optimization backbone is FedProx,

$(\text{layer}, \text{channel})$13

with the proximal term applied only to the currently trained head. Server-side aggregation is role-specific. For a parameter $(\text{layer}, \text{channel})$14, define $(\text{layer}, \text{channel})$15 and let $(\text{layer}, \text{channel})$16 be the clients owning that role. If $(\text{layer}, \text{channel})$17, the old global parameter is kept: $(\text{layer}, \text{channel})$18 Otherwise,

$(\text{layer}, \text{channel})$19

The unnormalized weights are role dependent: $(\text{layer}, \text{channel})$20

$(\text{layer}, \text{channel})$21

followed by normalization (Wang et al., 26 Sep 2025).

The rationale is explicit. The paper states that it is difficult to aggregate multiple types of data with federated learning alone because embedding and semantic spaces differ fundamentally, which can cause conflicts during model aggregation. The role-aware strategy is intended to isolate “aggregation conflicts and convergence oscillations caused by different embeddings,” avoid “cross-modal gradient interference,” and allow any client to invoke any modality head trained elsewhere at inference time. Relative to FedMM, the method federates branch heads and fusion modules at the prediction layer rather than feature extractors and global prototypes (Wang et al., 26 Sep 2025).

The empirical results reported for this role-aware implementation are strong but should be interpreted with the paper’s own caveats. On TR-OP, the Fusion head reaches accuracy $(\text{layer}, \text{channel})$22 with FPR $(\text{layer}, \text{channel})$23 across two data types. On the image subset ablation it attains accuracy $(\text{layer}, \text{channel})$24 with FPR $(\text{layer}, \text{channel})$25. On WebPhish HTML it obtains accuracy $(\text{layer}, \text{channel})$26 and FPR $(\text{layer}, \text{channel})$27; on TR-OP raw HTML, accuracy $(\text{layer}, \text{channel})$28 and FPR $(\text{layer}, \text{channel})$29. The paper repeatedly attributes stable convergence to the bucketed, hard-gated design, but it does not include a direct controlled ablation against plain FedProx without bucketing or ordinary FedAvg on the same architecture. It also notes that equal weighting in the HTML bucket slightly slows the convergence of the low-data TR-OP client, that known modality labels are required, and that if both HTML and Fusion exist in the client, the HTML head or image head will be updated twice during backpropagation (Wang et al., 26 Sep 2025).

7. Recurring methodological lessons and limits

A recurring lesson across these papers is that bucket recovery and role identification are distinct operations. The Mamba-2 work shows that a probe-derived signature may identify both a detection layer and an execution layer, and that representation-level statistics such as class-conditional cosine or even near-perfect linear decodability do not determine functional role (Jiang, 30 May 2026). The semantic-layer work shows that producing the correct total requires explicit normalization over join-key groups, not heuristic deduplication (Huang et al., 2023). Dynamic grouping shows that a target bucket can remain fixed while the evidence bucket expands, which separates interpretive identity from computational support (Loo, 2024). Provenance theory shows that aggregate values themselves may need internal contributor structure rather than a single tuple-level annotation (Amsterdamer et al., 2011).

Another lesson is that the presence of buckets does not by itself imply role-awareness. In ranking problems, bucket orders are powerful summaries, but the papers do not model explicit roles, observed role labels, or role-conditioned fairness constraints (Achab et al., 2018, Aledo et al., 19 Feb 2025). In provenance, GROUP BY provides a bucket analogue, but there is no first-class role dimension unless roles are encoded in annotations, attributes, or the value monoid (Amsterdamer et al., 2011). In the semantic-layer and dynamic-grouping papers, the transfer to role-aware bucket aggregation is conceptually natural, but the role semantics remain an interpretation rather than a named formal object (Huang et al., 2023, Loo, 2024).

The most important misconception rejected by this literature is that aggregation errors are always quantitative rather than semantic. The cited works argue the opposite. A bucket can preserve or destroy the very property one hopes to infer: causal execution in mechanistic interpretability, total-mass preservation in BI systems, support adequacy in sparse grouped estimation, or modality coherence in federated optimization. This suggests that bucket design should be evaluated by invariants that match the intended use: class-conditional ablation for functional role in Mamba-2, per-group weight normalization for metric consistency, support predicates for dynamic grouping, provenance expressions for contributor tracing, and role-compatible client selection for multimodal federation (Jiang, 30 May 2026, Huang et al., 2023, Loo, 2024, Amsterdamer et al., 2011, Wang et al., 26 Sep 2025).

The resulting concept of role-aware bucket aggregation is therefore best understood as a methodological family rather than a single architecture. In the strongest explicit case, it means averaging only the parameters associated with an activated expert role. In the strongest implicit case, it means that probe-defined buckets must be separated by causal role rather than by representational resemblance alone. More generally, it denotes aggregation schemes in which bucket boundaries are chosen so that the aggregate remains faithful to the role structure present in the data or model.