{"paper":{"title":"$\\phi$-Balancing for Mixture-of-Experts Training","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Mixture-of-experts models achieve population-level expert balance by minimizing a strictly convex potential of the expected routing distribution.","cross_cats":["math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Chen Liang, Jonathan Li, Lizhang Chen, Ni Lao, Qiang Liu, Qi Wang, Runlong Liao, Shuozhe Li","submitted_at":"2026-05-14T20:39:28Z","abstract_excerpt":"Mixture-of-Experts (MoE) models rely on balanced expert utilization to fully realize their scalability. However, existing load-balancing methods are largely heuristic and operate on noisy mini-batch assignment statistics, introducing bias relative to population-level objectives. We propose $\\phi$-balancing, a principled framework that directly targets population-level expert balance by minimizing a strictly convex, symmetric, and differentiable potential of the expected routing distribution. Using convex duality, we derive an equivalent min-max formulation and obtain a simple online algorithm "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Across large-scale pretraining and downstream fine-tuning, φ-balancing consistently outperforms prior Switch-style and loss-free baselines, demonstrating more stable and effective expert utilization.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That minimizing the chosen strictly convex potential of the expected routing distribution produces the desired population-level balance and that the EMA-based online approximation via mirror descent faithfully tracks the population objective without introducing new bias (abstract, paragraph on derivation).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"φ-balancing is a convex optimization method for population-level expert balance in MoE training that derives an online EMA adjustment and outperforms heuristic baselines.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Mixture-of-experts models achieve population-level expert balance by minimizing a strictly convex potential of the expected routing distribution.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"955ebbb2ec6dbf24025c7adfc1f83c49aa3ea386480fab272d414054421ae24d"},"source":{"id":"2605.15403","kind":"arxiv","version":1},"verdict":{"id":"0a675f24-6592-4b5c-8477-cb87f4cc71cd","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T16:11:46.193593Z","strongest_claim":"Across large-scale pretraining and downstream fine-tuning, φ-balancing consistently outperforms prior Switch-style and loss-free baselines, demonstrating more stable and effective expert utilization.","one_line_summary":"φ-balancing is a convex optimization method for population-level expert balance in MoE training that derives an online EMA adjustment and outperforms heuristic baselines.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That minimizing the chosen strictly convex potential of the expected routing distribution produces the desired population-level balance and that the EMA-based online approximation via mirror descent faithfully tracks the population objective without introducing new bias (abstract, paragraph on derivation).","pith_extraction_headline":"Mixture-of-experts models achieve population-level expert balance by minimizing a strictly convex potential of the expected routing distribution."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15403/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T16:31:18.268850Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:25:54.221458Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T16:23:51.333105Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T15:50:50.932555Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.158666Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.715382Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"ac86f12ecb5a351f2b55165686c2a843bd71bf0d5f5c8899befc32acb3aaf78f"},"references":{"count":49,"sample":[{"doi":"","year":2024,"title":"Lion secretly solves a constrained optimization: As Lyapunov predicts","work_id":"86bf0e47-3d82-47e0-ad43-dbf2d3122d34","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Evaluating Large Language Models Trained on Code","work_id":"042493e9-b26f-4b4e-bbde-382072ca9b08","ref_index":2,"cited_arxiv_id":"2107.03374","is_internal_anchor":true},{"doi":"","year":2023,"title":"Chen, X., Liang, C., Huang, D., Real, E., Wang, K., Pham, H., Dong, X., Luong, T., Hsieh, C., Lu, Y ., and Le, Q. 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