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

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Computer Science > Machine Learning

arXiv:2607.23390 (cs)
[Submitted on 25 Jul 2026]

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

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Abstract: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 $D\rho_z$ 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.
Comments: 141 pages, 26 figures, 15 tables. Includes complete proofs and documents the QReplace decision-support and Lean 4 verification companions. To be submitted to the Journal of Machine Learning Research
Subjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
Cite as: arXiv:2607.23390 [cs.LG]
  (or arXiv:2607.23390v1 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.2607.23390
arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Mojtaba Soltanalian [view email]
[v1] Sat, 25 Jul 2026 23:29:23 UTC (11,201 KB)
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