{"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/pde-based-optimal-strategy-for-unconstrained","title":"PDE-Based Optimal Strategy for Unconstrained Online Learning","arxiv_id":"2201.07877","date":"2022-01-19","proceeding":null,"authors":["ZhiYu Zhang","Ashok Cutkosky","Ioannis Paschalidis"],"abstract":"Unconstrained Online Linear Optimization (OLO) is a practical problem setting to study the training of machine learning models. Existing works proposed a number of potential-based algorithms, but in general the design of these potential functions relies heavily on guessing. To streamline this workflow, we present a framework that generates new potential functions by solving a Partial Differential Equation (PDE). Specifically, when losses are 1-Lipschitz, our framework produces a novel algorithm with anytime regret bound $C\\sqrt{T}+||u||\\sqrt{2T}[\\sqrt{\\log(1+||u||/C)}+2]$, where $C$ is a user-specified constant and $u$ is any comparator unknown and unbounded a priori. Such a bound attains an optimal loss-regret trade-off without the impractical doubling trick. Moreover, a matching lower bound shows that the leading order term, including the constant multiplier $\\sqrt{2}$, is tight. To our knowledge, the proposed algorithm is the first to achieve such optimalities.","url_abs":"https://arxiv.org/abs/2201.07877v2","url_pdf":"https://arxiv.org/pdf/2201.07877v2.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":"pde-based-optimal-strategy-for-unconstrained","repo_url":"https://github.com/zhiyuzz/icml2022-pde-potential","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[{"method_slug":"hoc","method_name":"HOC"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2201.07877","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2201.07877"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+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/zhiyuzz/icml2022-pde-potential","reach":null}],"summary":{"ran":1,"unverified":1},"by_repo_kind":{"official":{"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":2,"samples":[{"code_sha256_prefix":"1ce01fede64c59b2","entry":"OneDimPositive","repo":"zhiyuzz/icml2022-pde-potential","repo_kind":"official","path":"Algorithms/Algorithm_1d.py","file_url":"https://github.com/zhiyuzz/icml2022-pde-potential/blob/HEAD/Algorithms/Algorithm_1d.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"1ce01fede64c59b2"}},{"code_sha256_prefix":"735b09992bf5d6d9","entry":"potential_conjugate","repo":"zhiyuzz/icml2022-pde-potential","repo_kind":"official","path":"Algorithms/Algorithm_1d.py","file_url":"https://github.com/zhiyuzz/icml2022-pde-potential/blob/HEAD/Algorithms/Algorithm_1d.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":"HONOURS","metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"735b09992bf5d6d9"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}