House-monotone multi-level apportionment has logarithmic quota discrepancy
arXiv:2608.02559
2026
Architecture
2 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper gives a constructive discrepancy theory for hierarchical, progressive integer allocation: a single monotone sequence of one-hot choices can incur logarithmic error even though every individual house size has a quota-feasible allocation. The transferable asset is the static-versus-dynamic distinction, together with a binary van der Corput schedule whose hierarchical prefix error is explicitly bounded by O(log D). This suggests replacing random or greedy progressive categorical rounding with deterministic low-discrepancy schedules when a model must distribute activation, pruning, quantization, or update opportunities over time while preserving proportional usage at every prefix and across nested groups.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Replace random or greedy one-expert-at-a-time activation with a deterministic binary van der Corput sequence. At each training or inference step, the schedule chooses an expert whose cumulative usage remains close to its proportional target, while recursively balancing nested expert groups rather than only balancing individual experts.
Useful7/10
Difficulty5/10
Novelty7/10
Unverified
2026
Use the paper's monotone allocation viewpoint to decide which parameters receive progressively more pruning, bit reduction, or update opportunities. A van der Corput order prevents a fixed early prefix of structured groups from being systematically over-pruned, while the logarithmic comb bound quantifies worst cumulative imbalance across nested groups.
Useful6/10
Difficulty4/10
Novelty6/10