{"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/rethinking-parameter-counting-in-deep-models","title":"Rethinking Parameter Counting in Deep Models: Effective Dimensionality Revisited","arxiv_id":"2003.02139","date":"2020-03-04","proceeding":null,"authors":["Wesley J. Maddox","Gregory Benton","Andrew Gordon Wilson"],"abstract":"Neural networks appear to have mysterious generalization properties when using parameter counting as a proxy for complexity. Indeed, neural networks often have many more parameters than there are data points, yet still provide good generalization performance. Moreover, when we measure generalization as a function of parameters, we see double descent behaviour, where the test error decreases, increases, and then again decreases. We show that many of these properties become understandable when viewed through the lens of effective dimensionality, which measures the dimensionality of the parameter space determined by the data. We relate effective dimensionality to posterior contraction in Bayesian deep learning, model selection, width-depth tradeoffs, double descent, and functional diversity in loss surfaces, leading to a richer understanding of the interplay between parameters and functions in deep models. We also show that effective dimensionality compares favourably to alternative norm- and flatness- based generalization measures.","url_abs":"https://arxiv.org/abs/2003.02139v2","url_pdf":"https://arxiv.org/pdf/2003.02139v2.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":"rethinking-parameter-counting-in-deep-models","repo_url":"https://github.com/g-benton/hessian-eff-dim","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"Apache-2.0"}}],"tasks":[{"task_slug":"diversity","task_name":"Diversity"},{"task_slug":"model-selection","task_name":"Model Selection"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2003.02139","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2003.02139"}},"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/g-benton/hessian-eff-dim","reach":{"status":"ok","spdx":"Apache-2.0"}}],"summary":{"unverified":8},"by_repo_kind":{"official":{"samples":8,"ran":0,"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":"a164b6cd82f40a9f","entry":"ConvBNrelu","repo":"g-benton/hessian-eff-dim","repo_kind":"official","path":"hess/nets/cifar_net.py","file_url":"https://github.com/g-benton/hessian-eff-dim/blob/HEAD/hess/nets/cifar_net.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"Apache-2.0","inline_ok":true,"mcp_get_code":{"code_sha256":"a164b6cd82f40a9f"}},{"code_sha256_prefix":"e9d3b739701f64b2","entry":"block","repo":"g-benton/hessian-eff-dim","repo_kind":"official","path":"hess/nets/convnet.py","file_url":"https://github.com/g-benton/hessian-eff-dim/blob/HEAD/hess/nets/convnet.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"Apache-2.0","inline_ok":true,"mcp_get_code":{"code_sha256":"e9d3b739701f64b2"}},{"code_sha256_prefix":"873c3f3bc0248b85","entry":"dataloader_loss_surface","repo":"g-benton/hessian-eff-dim","repo_kind":"official","path":"hess/loss_surfaces/dataloader_loss_surface.py","file_url":"https://github.com/g-benton/hessian-eff-dim/blob/HEAD/hess/loss_surfaces/dataloader_loss_surface.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"Apache-2.0","inline_ok":true,"mcp_get_code":{"code_sha256":"873c3f3bc0248b85"}},{"code_sha256_prefix":"4192f983970465a5","entry":"get_loss_surface","repo":"g-benton/hessian-eff-dim","repo_kind":"official","path":"hess/loss_surfaces/loss_surfaces.py","file_url":"https://github.com/g-benton/hessian-eff-dim/blob/HEAD/hess/loss_surfaces/loss_surfaces.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"Apache-2.0","inline_ok":true,"mcp_get_code":{"code_sha256":"4192f983970465a5"}},{"code_sha256_prefix":"32bc1bd783d894e5","entry":"get_plane","repo":"g-benton/hessian-eff-dim","repo_kind":"official","path":"experiments/cifar-homogeneity/compute_loss_surface.py","file_url":"https://github.com/g-benton/hessian-eff-dim/blob/HEAD/experiments/cifar-homogeneity/compute_loss_surface.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"Apache-2.0","inline_ok":true,"mcp_get_code":{"code_sha256":"32bc1bd783d894e5"}},{"code_sha256_prefix":"206350ffb5317c91","entry":"get_plane","repo":"g-benton/hessian-eff-dim","repo_kind":"official","path":"hess/loss_surfaces/loss_surfaces.py","file_url":"https://github.com/g-benton/hessian-eff-dim/blob/HEAD/hess/loss_surfaces/loss_surfaces.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"Apache-2.0","inline_ok":true,"mcp_get_code":{"code_sha256":"206350ffb5317c91"}},{"code_sha256_prefix":"7498ce072ecf4350","entry":"gram_schmidt","repo":"g-benton/hessian-eff-dim","repo_kind":"official","path":"experiments/cifar-loss-surfaces/loss_surface_runner.py","file_url":"https://github.com/g-benton/hessian-eff-dim/blob/HEAD/experiments/cifar-loss-surfaces/loss_surface_runner.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"Apache-2.0","inline_ok":true,"mcp_get_code":{"code_sha256":"7498ce072ecf4350"}},{"code_sha256_prefix":"4806d38b109f68f5","entry":"loss_getter","repo":"g-benton/hessian-eff-dim","repo_kind":"official","path":"experiments/cifar-homogeneity/compute_loss_surface.py","file_url":"https://github.com/g-benton/hessian-eff-dim/blob/HEAD/experiments/cifar-homogeneity/compute_loss_surface.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"Apache-2.0","inline_ok":true,"mcp_get_code":{"code_sha256":"4806d38b109f68f5"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}