{"work":{"id":"663a7d08-8916-427b-bc7f-d58a68ab4780","openalex_id":"https://openalex.org/W4315588185","doi":"10.48550/arxiv.2301.02679","arxiv_id":"2301.02679","raw_key":null,"title":"Grokking modular arithmetic","authors":null,"authors_text":"Grokking modular arithmetic , author=","year":2023,"venue":"cs.LG","abstract":"We present a simple neural network that can learn modular arithmetic tasks and exhibits a sudden jump in generalization known as ``grokking''. Concretely, we present (i) fully-connected two-layer networks that exhibit grokking on various modular arithmetic tasks under vanilla gradient descent with the MSE loss function in the absence of any regularization; (ii) evidence that grokking modular arithmetic corresponds to learning specific feature maps whose structure is determined by the task; (iii) analytic expressions for the weights -- and thus for the feature maps -- that solve a large class of modular arithmetic tasks; and (iv) evidence that these feature maps are also found by vanilla gradient descent as well as AdamW, thereby establishing complete interpretability of the representations learnt by the network.","external_url":"https://arxiv.org/abs/2301.02679","cited_by_count":3,"metadata_source":"pith","metadata_fetched_at":"2026-08-05T02:28:24.338817+00:00","pith_arxiv_id":"2301.02679","created_at":"2026-05-10T00:49:48.450251+00:00","updated_at":"2026-08-05T02:28:24.338817+00:00","title_quality_ok":false,"display_title":"arXiv preprint arXiv:2301.02679 , year=","render_title":"arXiv preprint arXiv:2301.02679 , year="},"hub":{"state":{"work_id":"663a7d08-8916-427b-bc7f-d58a68ab4780","tier":"hub","tier_reason":"10+ Pith inbound or 1,000+ external citations","pith_inbound_count":18,"external_cited_by_count":3,"distinct_field_count":5,"first_pith_cited_at":"2024-08-23T09:33:48+00:00","last_pith_cited_at":"2026-07-08T06:49:48+00:00","author_build_status":"not_needed","summary_status":"needed","contexts_status":"needed","graph_status":"needed","ask_index_status":"not_needed","reader_status":"not_needed","recognition_status":"not_needed","updated_at":"2026-08-22T22:19:41.424975+00:00","tier_text":"hub"},"tier":"hub","role_counts":[{"context_role":"background","n":3}],"polarity_counts":[{"context_polarity":"background","n":2},{"context_polarity":"support","n":1}],"runs":{},"summary":{},"graph":{},"authors":[]}}