Optimal Practice Allocation Under Learning Saturation
Abstract
A saying attributed to Bruce Lee unfavorably contrasts a martial artist who has practiced ten thousand kicks once each with another who has practiced a single kick ten thousand times. We read the saying as implicitly raising a question about how to allocate a fixed practice budget among several skills, and we show that the answer is governed by two ingredients: the shape of the learning curve that converts practice into skill, and the rule by which separate skills are aggregated into overall effectiveness. Neither ingredient alone settles the matter. Our central result reduces the multivariable allocation problem to the maximization of a single scalar \emph{efficiency function} eff(x)=f(x)p/x; under a simple uniqueness and divisibility condition, every optimal practice schedule is \emph{balanced}, dividing the budget equally among a definite number of skills. The optimal number of skills is then determined by a transparent criterion equating the elasticity of the learning curve to the reciprocal of the aggregation parameter. Hard-saturation models, heterogeneous learning rates, a sharp specialization threshold, and a genuinely intermediate optimal repertoire all follow as consequences.
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BibTeX
@misc{rao2026optimal,
title = {{Optimal Practice Allocation Under Learning Saturation}},
author = {Shrisha Rao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05501},
eprint = {2609.05501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05501},
}
RIS
TY - UNPB
TI - Optimal Practice Allocation Under Learning Saturation
AU - Shrisha Rao
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.05501
UR - https://arxiv.org/abs/2609.05501
AB - A saying attributed to Bruce Lee unfavorably contrasts a martial artist who has practiced ten thousand kicks once each with another who has practiced a single kick ten thousand times. We read the saying as implicitly raising a question about how to allocate a fixed practice budget among several skills, and we show that the answer is governed by two ingredients: the shape of the learning curve that converts practice into skill, and the rule by which separate skills are aggregated into overall effectiveness. Neither ingredient alone settles the matter. Our central result reduces the multivariable allocation problem to the maximization of a single scalar \emph{efficiency function} eff(x)=f(x)^p/x; under a simple uniqueness and divisibility condition, every optimal practice schedule is \emph{balanced}, dividing the budget equally among a definite number of skills. The optimal number of skills is then determined by a transparent criterion equating the elasticity of the learning curve to the reciprocal of the aggregation parameter. Hard-saturation models, heterogeneous learning rates, a sharp specialization threshold, and a genuinely intermediate optimal repertoire all follow as consequences.
ER -