mlsanityData to 14 Oct 2026

When Can Depth Replace Precision? A Resource Theory of Quantized Neural Computation

arXiv 2607.23390 ↗First posted 25 Jul 2026cs.LG math.OCcited by 1

Not on the map. Why?

Abstract (arXiv)

When can additional low-bit residual computation replace missing numerical precision for a fixed input-output map? We model a quantized residual system over a fixed horizon as a pure schedule selecting fields from a declared low-bit operation library, and use relaxed controls to characterize its infinite-depth limit. The distance from the target to the closed relaxed reachable set is the exact structural floor: no increase in depth can remove it for that library. Pure schedules approach the relaxed class at rate O(D-1) under bounded-variation time dependence and O(D-vartheta+D-1) under Holder dependence of exponent vartheta. Execution arithmetic can reverse this conclusion: full-state write-back introduces a Dρz penalty and can freeze residual updates, whereas increment error feedback replaces this growth by a bounded carry term and obeys an exact common-lattice conservation law. A fixed-teacher converse makes this rate sharp: for coherent depth-L first-order high-precision comparators, accuracy matching requires D=Theta(L). Learned codebooks add a metadata resource, while state-dependent routing introduces hybrid event conditions. Verified primal and dual bounds yield feasible, impossible, or unresolved decisions before training. Companion software implements the workflow, and Lean 4 machine-checks the exact discrete core. Depth replaces precision only relative to a declared library, horizon, execution semantics, and routing model.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 31 of its 107 references that are in the map. First posted on arXiv 25 Jul 2026.

Builds on

31 of 107 references.

References · 31

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19 more
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Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Depth Laws for the Precision Floor of Trained Neural Networks: Amplification, Residual Scaling, and a Quantization-Aware Training Paradoxcited by 0

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  5. 2022Few-Bit Backward: Quantized Gradients of Activation Functions for Memory Footprint Reduction
  6. 2023A Practical Mixed Precision Algorithm for Post-Training Quantization
  7. 2025Pushing the Limits of Low-Bit Optimizers: A Focus on EMA Dynamics
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