{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/lagrangian-decomposition-for-neural-network","title":"Lagrangian Decomposition for Neural Network Verification","arxiv_id":"2002.10410","date":"2020-02-24","proceeding":null,"authors":["Rudy Bunel","Alessandro De Palma","Alban Desmaison","Krishnamurthy Dvijotham","Pushmeet Kohli","Philip H. S. Torr","M. Pawan Kumar"],"abstract":"A fundamental component of neural network verification is the computation of bounds on the values their outputs can take. Previous methods have either used off-the-shelf solvers, discarding the problem structure, or relaxed the problem even further, making the bounds unnecessarily loose. We propose a novel approach based on Lagrangian Decomposition. Our formulation admits an efficient supergradient ascent algorithm, as well as an improved proximal algorithm. Both the algorithms offer three advantages: (i) they yield bounds that are provably at least as tight as previous dual algorithms relying on Lagrangian relaxations; (ii) they are based on operations analogous to forward/backward pass of neural networks layers and are therefore easily parallelizable, amenable to GPU implementation and able to take advantage of the convolutional structure of problems; and (iii) they allow for anytime stopping while still providing valid bounds. Empirically, we show that we obtain bounds comparable with off-the-shelf solvers in a fraction of their running time, and obtain tighter bounds in the same time as previous dual algorithms. This results in an overall speed-up when employing the bounds for formal verification. Code for our algorithms is available at https://github.com/oval-group/decomposition-plnn-bounds.","url_abs":"https://arxiv.org/abs/2002.10410v3","url_pdf":"https://arxiv.org/pdf/2002.10410v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"lagrangian-decomposition-for-neural-network","repo_url":"https://github.com/oval-group/decomposition-plnn-bounds","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"lagrangian-decomposition-for-neural-network","repo_url":"https://github.com/oval-group/scaling-the-convex-barrier","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":null,"task_name":"GPU"},{"task_slug":null,"task_name":"valid"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2002.10410","atlas_url":"https://app.syntology.ai/?focus=2002.10410","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2002.10410"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/oval-group/scaling-the-convex-barrier","reach":null},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/oval-group/decomposition-plnn-bounds","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"ran":4,"ran_draft_wrong":1,"unverified":2},"by_repo_kind":{"official":{"samples":5,"ran":4,"repositories":1},"listed":{"samples":2,"ran":1,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":0,"samples":[{"code_sha256_prefix":"afab9dc2ebdd02aa","entry":"bdot","repo":"oval-group/decomposition-plnn-bounds","repo_kind":"official","path":"plnn_bounds/proxlp_solver/utils.py","file_url":"https://github.com/oval-group/decomposition-plnn-bounds/blob/HEAD/plnn_bounds/proxlp_solver/utils.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"afab9dc2ebdd02aa"}},{"code_sha256_prefix":"03249132743194cc","entry":"bl2_norm","repo":"oval-group/decomposition-plnn-bounds","repo_kind":"official","path":"plnn_bounds/proxlp_solver/utils.py","file_url":"https://github.com/oval-group/decomposition-plnn-bounds/blob/HEAD/plnn_bounds/proxlp_solver/utils.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"03249132743194cc"}},{"code_sha256_prefix":"c1b9258718b8fe48","entry":"make_elided_models","repo":"oval-group/scaling-the-convex-barrier","repo_kind":"listed","path":"tools/bounding_tools/anderson_cifar_bound_comparison.py","file_url":"https://github.com/oval-group/scaling-the-convex-barrier/blob/HEAD/tools/bounding_tools/anderson_cifar_bound_comparison.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":"deterministic","behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"c1b9258718b8fe48"}},{"code_sha256_prefix":"b4ad52a116df1451","entry":"opt_relu_pair","repo":"oval-group/decomposition-plnn-bounds","repo_kind":"official","path":"plnn_bounds/proxlp_solver/by_pairs.py","file_url":"https://github.com/oval-group/decomposition-plnn-bounds/blob/HEAD/plnn_bounds/proxlp_solver/by_pairs.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"b4ad52a116df1451"}},{"code_sha256_prefix":"1c49e5ae3f68bbc7","entry":"prod","repo":"oval-group/decomposition-plnn-bounds","repo_kind":"official","path":"plnn_bounds/proxlp_solver/utils.py","file_url":"https://github.com/oval-group/decomposition-plnn-bounds/blob/HEAD/plnn_bounds/proxlp_solver/utils.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"1c49e5ae3f68bbc7"}},{"code_sha256_prefix":"9980cffefb52e0f4","entry":"cifar_loaders","repo":"oval-group/scaling-the-convex-barrier","repo_kind":"listed","path":"tools/bounding_tools/anderson_cifar_bound_comparison.py","file_url":"https://github.com/oval-group/scaling-the-convex-barrier/blob/HEAD/tools/bounding_tools/anderson_cifar_bound_comparison.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"9980cffefb52e0f4"}},{"code_sha256_prefix":"298cf2a3ecfce374","entry":"opt_lin_domain","repo":"oval-group/decomposition-plnn-bounds","repo_kind":"official","path":"plnn_bounds/proxlp_solver/by_pairs.py","file_url":"https://github.com/oval-group/decomposition-plnn-bounds/blob/HEAD/plnn_bounds/proxlp_solver/by_pairs.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"298cf2a3ecfce374"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}