{"paper":{"title":"Parameterized Complexity of Stationarity Testing for Piecewise-Affine Functions and Shallow CNN Losses","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Testing approximate stationarity for continuous piecewise-affine functions is XP-tractable in fixed dimension for some cases but W[1]-hard for others, with ETH lower bounds excluding subexponential dependence on dimension.","cross_cats":["cs.CC","cs.LG"],"primary_cat":"math.OC","authors_text":"Yuhan Ye","submitted_at":"2026-05-11T08:59:57Z","abstract_excerpt":"We study the parameterized complexity of testing approximate first-order stationarity at a prescribed point for continuous piecewise-affine (PA) functions, a basic task in nonsmooth optimization. PA functions form a canonical model for nonsmooth stationarity testing and capture the local polyhedral geometry that appears in ReLU-type training losses. Recent work by Tian and So (SODA 2025) shows that testing approximate stationarity notions for PA functions is computationally intractable in the worst case, and identifies fixed-dimensional tractability as an open direction.\n  We address this dire"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give XP algorithms in fixed dimension for the tractable sides, and prove W[1]-hardness for the complementary sides. Moreover, lower bounds under the Exponential Time Hypothesis rule out algorithms running in time ρ(d) size^{o(d)} for any computable function ρ, where size denotes the total binary encoding length of the stationarity-testing instance.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that continuous piecewise-affine functions form a canonical model that captures the local polyhedral geometry appearing in ReLU-type training losses, and that the chosen notion of approximate first-order stationarity is the appropriate one for the parameterized analysis.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper fully characterizes the parameterized complexity of approximate first-order stationarity testing for continuous piecewise-affine functions and shallow ReLU CNN losses with respect to the dimension parameter, including XP algorithms, W[1]-hardness, and ETH lower bounds.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Testing approximate stationarity for continuous piecewise-affine functions is XP-tractable in fixed dimension for some cases but W[1]-hard for others, with ETH lower bounds excluding subexponential dependence on dimension.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"69b02a619c49e94e162790d813640136c937ac1346bfcabe496965fc1264e526"},"source":{"id":"2605.10219","kind":"arxiv","version":2},"verdict":{"id":"4c2af462-eacd-4573-855a-9f6cc16bff42","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T03:27:42.489870Z","strongest_claim":"We give XP algorithms in fixed dimension for the tractable sides, and prove W[1]-hardness for the complementary sides. Moreover, lower bounds under the Exponential Time Hypothesis rule out algorithms running in time ρ(d) size^{o(d)} for any computable function ρ, where size denotes the total binary encoding length of the stationarity-testing instance.","one_line_summary":"The paper fully characterizes the parameterized complexity of approximate first-order stationarity testing for continuous piecewise-affine functions and shallow ReLU CNN losses with respect to the dimension parameter, including XP algorithms, W[1]-hardness, and ETH lower bounds.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that continuous piecewise-affine functions form a canonical model that captures the local polyhedral geometry appearing in ReLU-type training losses, and that the chosen notion of approximate first-order stationarity is the appropriate one for the parameterized analysis.","pith_extraction_headline":"Testing approximate stationarity for continuous piecewise-affine functions is XP-tractable in fixed dimension for some cases but W[1]-hard for others, with ETH lower bounds excluding subexponential dependence on dimension."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.10219/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T06:22:00.902625Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T15:38:33.571346Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T11:31:19.856695Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T09:35:20.120462Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"6e14a490263ebbf3bf58b5d39f167e4ffce29708e31733b22957be4203a9d947"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"5361fb8c69c46fc15d663821864037ab52924898d611d23293f738c430f58298"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}