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We provide a theoretical analysis of DeepBern-Nets (DBNs) -- networks employing learnable Bernstein polynomial activations -- showing that their approximation error decays with the network depth $L$ and the polynomial order $n$ with a rate of $\\mathcal{O}(n^{-L})$, exponentially faster than the polynomial rate of ReLU architectures while remaining fully differentiable. We validate these predictions through $1{,}344$ experiments on large scientific datasets (HIGGS and SUSY), comparing DBNs against ReLU, Leaky ReLU, SELU, and GeLU. DBNs achieve over $70\\%$ parameter reduction across the majority of architectures -- reaching $99.9\\%$ at scale -- converge to ReLU's final loss in as few as $26\\%$ of the trainin","title":"Exponential Approximation Rates and Parameter Efficiency of Learnable Bernstein Activations","url":"https://arxiv.org/abs/2602.04264","vendor":"arxiv_cs_ai"},"summary":"arXiv:2602.04264v2 Announce Type: replace-cross \nAbstract: The choice of activation function fundamentally shapes the representational capacity and parameter efficiency of deep neural networks, yet most widely used activations lack rigorous theoretical guarantees on these properties. We provide a theoretical analysis of DeepBern-Nets (DBNs) -- networks employing learnable Bernstein polynomial activations -- showing that their approximation error decays with the network depth $L$ and the polynomial order $n$ with a rate of $\\mathcal{O}(n^{-L})$, exponentially faster than the polynomial rate of ReLU architectures while remaining fully differentiable. We validate these predictions through $1{,}344$ experiments on large scientific datasets (HIGGS and SUSY), comparing DBNs against ReLU, Leaky ReLU, SELU, and GeLU. DBNs achieve over $70\\%$ parameter reduction across the majority of architectures -- reaching $99.9\\%$ at scale -- converge to ReLU's final loss in as few as $26\\%$ of the trainin","title":"Exponential Approximation Rates and Parameter Efficiency of Learnable Bernstein Activations","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-14T04:43:36Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2602.04264"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:4d995079ed59a1466c8e5407800e72deeaf4eff9d9e5cd574e05207f32697a39f2722b28a95f2ea58c0f7ca092f4aaa1077b77e4a544e324c22d14b241f33902","signer":"crovia.substrate","subject":{"observed_at":"2026-05-14T04:43:36Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2602.04264"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"2a6fede548f022b1e9580953621783677d944156ef58f7ffc817aa283d2884b0","leaf_index":132789,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"6374f321f3908c9196e59f6a89fa8ea1c128049042ac40c7f960a94a44c88f52","side":"left"},{"sibling":"8ebe8c670d0224fb74a4cc82d09b1d83afe074f9a52a823ac00c4712cc184612","side":"right"},{"sibling":"0ceed2313386f952d70aa1174fb539f8136dd05cb6e44648b077d86349cbe20d","side":"left"},{"sibling":"66e6b32447da7cd6a914e326b4a746bbb1256547739327d3b8228e0791fbc84c","side":"right"},{"sibling":"4c3d66db1f93ba8d22aead8f3b98163b53c43faaa0b5854f698d4797a7f1aa65","side":"left"},{"sibling":"695a4c4d44cd61cced2c94a1e621686058c017437bf0fdcf86516ffa6e66c772","side":"left"},{"sibling":"082bee6702d02c968f73550f57d052606dc8732cddb9a8af38542747b47966a4","side":"right"},{"sibling":"d390ceb521d99fbee195843b8c087d5b6d6b3c13cc9f5d0b36a1c937a84cb77a","side":"left"},{"sibling":"2ae8cbb1d93652ee36f693c3d63e733765fbafcd7765d6d596692bf393ce0a1d","side":"right"},{"sibling":"a957418f640d5dc3181a7628c2646bb86c6da0ea6888670b451e534693a0c7cb","side":"left"},{"sibling":"c03f0a468f574a08ffe8b17e1a17bd88216e1359f0e33a54447e160cd8675da0","side":"left"},{"sibling":"038ff12da6f55509125ef0d96e1e57dda29a2fe63bf03fba2af4cf7cbcd88b36","side":"right"},{"sibling":"b0419206fe62ef216df3900ca93cffd44df267435a4643dda354c1310079cf91","side":"right"},{"sibling":"0d4a9c03674f9d0ce64df15c15e9f656a54b41f93c428aac8e615fda26291956","side":"right"},{"sibling":"7856d920f3f1f1d2194c1ed7351bf0d674440df3cb423a6911f89cb3e9578c0b","side":"right"},{"sibling":"6ac6396bdd2e9df315427a46155531476e7f9012b4bd962e0d2d6d1209b11723","side":"right"},{"sibling":"7c8dc85cbfe43e19ac759ad176cfa11dba2467ae17927c471d5c55663c4f490d","side":"right"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_32e82fda1feef45d48efd6ac2b58fecc4beffb332d302eac444f949dd0264869"}}