{"paper":{"title":"Min-Max Optimization Requires Exponentially Many Queries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Any algorithm finding an ε-approximate stationary point in nonconvex-nonconcave min-max optimization requires exponentially many queries.","cross_cats":["cs.CC","cs.GT","cs.LG","math.OC"],"primary_cat":"cs.DS","authors_text":"Alexandros Hollender, Andrea Celli, Martino Bernasconi, Matteo Castiglioni","submitted_at":"2026-05-13T17:34:24Z","abstract_excerpt":"We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \\times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\\nabla f$, any algorithm that finds an $\\varepsilon$-approximate stationary point must make a number of queries that is exponential in $1/\\varepsilon$ or $d$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"any algorithm that finds an ε-approximate stationary point must make a number of queries that is exponential in 1/ε or d","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The function belongs to the nonconvex-nonconcave class and the oracle model provides exact access to f and ∇f; if the class is restricted further or oracles are noisy, the exponential bound may not apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Finding an ε-approximate stationary point for nonconvex-nonconcave min-max optimization over [0,1]^d × [0,1]^d requires exponentially many queries in 1/ε or d.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Any algorithm finding an ε-approximate stationary point in nonconvex-nonconcave min-max optimization requires exponentially many queries.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ffda400fef57ede54f7fc7ef63d26a17aabc4e4a165425de83f5282bc9125f1b"},"source":{"id":"2605.13806","kind":"arxiv","version":1},"verdict":{"id":"94b66018-a77a-46f8-927a-4cc78910af11","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:27:58.368989Z","strongest_claim":"any algorithm that finds an ε-approximate stationary point must make a number of queries that is exponential in 1/ε or d","one_line_summary":"Finding an ε-approximate stationary point for nonconvex-nonconcave min-max optimization over [0,1]^d × [0,1]^d requires exponentially many queries in 1/ε or d.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The function belongs to the nonconvex-nonconcave class and the oracle model provides exact access to f and ∇f; if the class is restricted further or oracles are noisy, the exponential bound may not apply.","pith_extraction_headline":"Any algorithm finding an ε-approximate stationary point in nonconvex-nonconcave min-max optimization requires exponentially many queries."},"references":{"count":6,"sample":[{"doi":"","year":1983,"title":"A polynomial-time algorithm for variational inequalities under the Minty condition","work_id":"15f567c2-0b71-4c5e-b450-d81bbd4b8c1e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"On the Role of Constraints in the Complexity of Min-Max Optimization","work_id":"764b540a-8226-4dc9-90a0-88e6e48c9cb6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Pure- Circuit: Tight Inapproximability for PPAD","work_id":"79a85b37-7c17-4421-9d69-e633baa8a8cd","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Playing large games using simple strategies","work_id":"6eb220a3-ad25-49e3-ab05-f828d9a6495d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Cycles in adversarial regularized learning","work_id":"4af6b150-6e68-476b-bbff-d1aca0d4a4c1","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":6,"snapshot_sha256":"5606ff24adf91320a307627312afd1b7e5ae8776ac14d3020a4fd5db84e41ce3","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"60bf2eb6e3c52714a90b37c9a8d03ab603701e62cfb5a90b28d142ec55f949a0"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}