Pith Number

pith:EK67N2VP

pith:2026:EK67N2VPWYA4PXRTDKUU3DPL3D

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Regret Equals Covariance: A Closed-Form Characterization for Stochastic Optimization

Irene Aldridge

Expected regret equals the covariance between uncertain costs and optimal decisions in stochastic optimization.

arxiv:2605.14019 v1 · 2026-05-13 · econ.EM · cs.LG · math.ST · stat.CO · stat.TH

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Claims

C1strongest claim

We prove that expected regret in any stochastic optimization problem admits the exact decomposition Regret(c) = Cov(c, π*(c)) + R(c), and for linear programs and unconstrained quadratic programs R(c)=0 exactly.

C2weakest assumption

The decomposition and exact equality for LPs and QPs rely on the problem being linear or unconstrained quadratic; the residual bound requires Lipschitz continuity, smoothness, and strong convexity of the objective.

C3one line summary

Expected regret equals covariance between costs and optimal decisions for linear and quadratic stochastic programs, with explicit bounds on the residual.

References

24 extracted · 24 resolved · 1 Pith anchors

[1] Differentiable convex optimization layers, 2019

[2] Generalization bounds in the predict-then-optimize framework, 2022

[3] Separability and decomposition in SAA for stochastic mixed- integer programming, 2015

[4] Machine learning for combinatorial optimization: A methodological tour d’horizon, 2021

[5] Statistical analysis of Wasserstein distributionally robust estimators, 2023

Cited by

2 papers in Pith

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Receipt and verification

First computed	2026-05-17T23:39:12.963622Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

22bdf6eaafb601c7de331aa94d8debd8c18c2980753b3c6a1f32734bed764051

Aliases

arxiv: 2605.14019 · arxiv_version: 2605.14019v1 · doi: 10.48550/arxiv.2605.14019 · pith_short_12: EK67N2VPWYA4 · pith_short_16: EK67N2VPWYA4PXRT · pith_short_8: EK67N2VP

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/EK67N2VPWYA4PXRTDKUU3DPL3D \
  | 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: 22bdf6eaafb601c7de331aa94d8debd8c18c2980753b3c6a1f32734bed764051

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "825690936dcc15da082aff09e668154c8c7b125a4f80f7b56d0fd429ef1675df",
    "cross_cats_sorted": [
      "cs.LG",
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      "stat.CO",
      "stat.TH"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "econ.EM",
    "submitted_at": "2026-05-13T18:32:44Z",
    "title_canon_sha256": "c3d3c3879a5b4d5f7770135eb611cca200b9ff60cb4c5e50e31f1a1b3bd94cb8"
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    "kind": "arxiv",
    "version": 1
  }
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