Pith Number

pith:OZUPR4BV

pith:2026:OZUPR4BVTWDMYSSOE6XPJ2ALWW

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$\phi$-Balancing for Mixture-of-Experts Training

Chen Liang, Jonathan Li, Lizhang Chen, Ni Lao, Qiang Liu, Qi Wang, Runlong Liao, Shuozhe Li

Mixture-of-experts models achieve population-level expert balance by minimizing a strictly convex potential of the expected routing distribution.

arxiv:2605.15403 v1 · 2026-05-14 · cs.LG · math.OC · stat.ML

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3 Author claim open · sign in to claim

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest 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.

C2weakest 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).

C3one 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.

References

49 extracted · 49 resolved · 9 Pith anchors

[1] Lion secretly solves a constrained optimization: As Lyapunov predicts 2024

[2] Evaluating Large Language Models Trained on Code 2026 · arXiv:2107.03374

[3] Chen, X., Liang, C., Huang, D., Real, E., Wang, K., Pham, H., Dong, X., Luong, T., Hsieh, C., Lu, Y ., and Le, Q. V . Symbolic discovery of optimization algorithms. InAd- vances in Neural Information 2023

[4] Ermoe: Eigen- reparameterized mixture-of-experts for stable routing and interpretable specialization

[5] Training Verifiers to Solve Math Word Problems · arXiv:2110.14168

Formal links

3 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:56.875852Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7668f8f0359d86cc4a4e27aef4e80bb5ba698b537c1265cabd5e7de8bbcb095f

Aliases

arxiv: 2605.15403 · arxiv_version: 2605.15403v1 · doi: 10.48550/arxiv.2605.15403 · pith_short_12: OZUPR4BVTWDM · pith_short_16: OZUPR4BVTWDMYSSO · pith_short_8: OZUPR4BV

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/OZUPR4BVTWDMYSSOE6XPJ2ALWW \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 7668f8f0359d86cc4a4e27aef4e80bb5ba698b537c1265cabd5e7de8bbcb095f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "4fc87225d0bf537713fef270d3fb111cfc24792dc12e6e3522fcc78d008b868f",
    "cross_cats_sorted": [
      "math.OC",
      "stat.ML"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-14T20:39:28Z",
    "title_canon_sha256": "3562498ca70109c3ba26eca3a5d2c962940053ae63d5d47f750bc534399aff91"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15403",
    "kind": "arxiv",
    "version": 1
  }
}