{"paper":{"title":"Regret Equals Covariance: A Closed-Form Characterization for Stochastic Optimization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Expected regret equals the covariance between uncertain costs and optimal decisions in stochastic optimization.","cross_cats":["cs.LG","math.ST","stat.CO","stat.TH"],"primary_cat":"econ.EM","authors_text":"Irene Aldridge","submitted_at":"2026-05-13T18:32:44Z","abstract_excerpt":"Regret is the cost of uncertainty in algorithmic decision-making. Quantifying regret typically requires computationally expensive simulation via Sample Average Approximation (SAA), with complexity $\\mathcal{O}(Bn^{2}d^{3})$ in the number of scenarios $B$, variables $n$, and constraints $d$. % This paper proves that expected regret in any stochastic optimization problem admits the exact decomposition % \\begin{equation*}\n  \\mathrm{Regret}(c)\n  = \\mathrm{Cov}(c,\\,\\pi^{*}(c)) + R(c), \\end{equation*} % where $c$ is the vector of uncertain parameters, $\\pi^{*}(c)$ is the optimal decision, and $R(c)$"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Expected regret equals covariance between costs and optimal decisions for linear and quadratic stochastic programs, with explicit bounds on the residual.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Expected regret equals the covariance between uncertain costs and optimal decisions in stochastic optimization.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9713cf1fff1ed15104bd082d95508f161a402188ef0f671b921201cf801e1c1c"},"source":{"id":"2605.14019","kind":"arxiv","version":1},"verdict":{"id":"9ab9cd1b-e234-484b-bcb2-01021a7e7220","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:30:51.911619Z","strongest_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.","one_line_summary":"Expected regret equals covariance between costs and optimal decisions for linear and quadratic stochastic programs, with explicit bounds on the residual.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"Expected regret equals the covariance between uncertain costs and optimal decisions in stochastic optimization."},"references":{"count":24,"sample":[{"doi":"","year":2019,"title":"Differentiable convex optimization layers,","work_id":"0b8d41ab-b7be-4bfe-950b-512d01b2e317","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Generalization bounds in the predict-then-optimize framework,","work_id":"d1c9bd2a-2684-4ad3-b2e0-a0fa5b88b29d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Separability and decomposition in SAA for stochastic mixed- integer programming,","work_id":"49b8cd66-0592-4030-bef1-e06c44a5c81e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Machine learning for combinatorial optimization: A methodological tour d’horizon,","work_id":"e1c9bf4d-732d-433d-8338-3620f3ffc46f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Statistical analysis of Wasserstein distributionally robust estimators,","work_id":"298cca64-22dd-4bea-bc92-8b8b9235d202","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":24,"snapshot_sha256":"39acbb6a8fd6c02781a8e1cdb4779f666f570e61954fcc02e8b7e3535fb4daa9","internal_anchors":1},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}