Pith Number

pith:L7J4X3YR

pith:2026:L7J4X3YRCMD5XL6VEMZ7KNTNS4

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Matrix-Decoupled Concentration for Autoregressive Sequences: Dimension-Free Guarantees for Sparse Long-Context Rewards

Pei-Sen Li

Matrix decoupling of causal dependencies yields dimension-free O(1) variance bounds for sparse rewards in autoregressive sequences.

arxiv:2605.06017 v2 · 2026-05-07 · cs.LG · math.PR

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Claims

C1strongest claim

we establish a sharp McDiarmid-type inequality for dependent sequences, governed strictly by the exact matrix-vector multiplication of the causal dependency resolvent and the target sensitivity vector. This Matrix-Decoupled Concentration (MDC) framework natively recovers optimal constants for Markov chains and exploits directed d-separation to yield order-optimal bounds for causal trees. Crucially, by exactly preserving the coordinate-wise sparsity of rewards within a strictly causal framework, MDC mathematically prevents scalar collapse, guaranteeing a dimension-free O(1) variance proxy

C2weakest assumption

The process admits a strictly causal filtration whose dependency structure can be represented by a well-defined resolvent matrix that exactly decouples the target sensitivity vector while preserving coordinate-wise sparsity of the reward.

C3one line summary

Matrix-Decoupled Concentration achieves dimension-free O(1) variance bounds for sparse rewards in strictly causal autoregressive sequences by decoupling via the exact matrix-vector product of the dependency resolvent and sensitivity vector.

Receipt and verification

First computed	2026-05-20T00:03:14.397270Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5fd3cbef111307dbafd52333f5366d9703651ef387605453fd0f7b08b9b136b0

Aliases

arxiv: 2605.06017 · arxiv_version: 2605.06017v2 · doi: 10.48550/arxiv.2605.06017 · pith_short_12: L7J4X3YRCMD5 · pith_short_16: L7J4X3YRCMD5XL6V · pith_short_8: L7J4X3YR

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/L7J4X3YRCMD5XL6VEMZ7KNTNS4 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 5fd3cbef111307dbafd52333f5366d9703651ef387605453fd0f7b08b9b136b0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a094d5c80fd5286647a73ce97f06c953cf00e0c65e4845ad8f29780b521ab35d",
    "cross_cats_sorted": [
      "math.PR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-07T11:12:59Z",
    "title_canon_sha256": "3c943ca4d616f4547e3e7f674fb2b2d2d87fbf47e53ca5cc2ad2ca252ff6a284"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.06017",
    "kind": "arxiv",
    "version": 2
  }
}