学习饱和下的最优练习分配
Optimal Practice Allocation Under Learning Saturation
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中文总结 AI 辅助
本文基于李小龙谚语提出固定练习预算在多项技能间的最优分配问题，证明答案取决于学习曲线形状与技能聚合规则，并将问题简化为效率函数最大化，得出平衡分配与最优技能数量准则。
中文摘要 AI 辅助
一句据称出自李小龙的谚语，将一位每种招式各练习一万次的武术家与另一位将单一招式练习一万次的武术家进行对比，并认为前者不如后者。我们将这句谚语解读为隐含地提出了一个问题：如何在多种技能之间分配固定的练习预算。我们证明，答案由两个要素决定：将练习转化为技能的学习曲线的形状，以及将不同技能聚合为整体效能的规则。这两个要素单独均无法决定答案。我们的核心结果将多变量分配问题简化为对单个标量“效率函数”\\(\eff(x)=f(x)^{p}/x\\) 的最大化；在简单的唯一性和可分割性条件下，每个最优练习计划都是“平衡的”，即将预算平均分配给确定数量的技能。最优技能数量随后由一个透明准则确定，该准则将学习曲线的弹性等同于聚合参数的倒数。硬饱和模型、异质学习速率、尖锐的专业化阈值以及真正介于两者之间的最优技能组合，均作为推论而得出。
英文摘要
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.
