可学习的预测不一定是可操作的：在线购买的适当修复维度
Learnable Predictions Need Not Be Actionable: Proper Repair Dimension for Online Buying
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中文总结 AI 辅助
研究在线购买中可学习性与可操作性差异，定义适当修复维度PRD(H)，证明相关不等式并给出示例，构建可操作购买实例，揭示Littlestone维度与实际成本关系，还给出组件化可操作预测类的正转移定理。
中文摘要 AI 辅助
可学习性和可操作性是在线预测的不同要求。 Littlestone维度控制普通在线可学习性，而适当修复维度控制可操作的在线购买。一个概念类在标准错误界意义上可能易于学习，但一旦要求算法保持适当并在线购买已实现的操作，作为实时可操作预测可能仍难以维护。对于有限二元概念类，本文通过版本空间上的动态规划定义了适当修复维度PRD(H)。PRD(H)的值恰好是必须通过每个可实现标记序列保持实时假设的适当学习者的最优确定性最坏情况修复次数。本文证明了Ldim(H) <= PRD(H) <= |H|-1，并给出了紧密示例：d维上的全类有Ldim = PRD = d，而通用坐标类U_n有Ldim(U_n) = floor(log_2 n) 且PRD(U_n) = n-1。这种差距直接转移到在线购买中。本文从每个适当类H构建了一个单位成本可操作购买实例，并证明终端基准是uOPT = 1，而每个确定性适当可操作算法在最坏情况下恰好支付1 + PRD(H)。对于U_{2^d}，尽管Littlestone维度为d，但确定性成本为2^d。本文还给出了组件化可操作预测类的正转移定理，表明有界修复维度与有界PRD负载拥塞会产生可控实际购买成本。
英文摘要
Learnability and actionability are different requirements for online predictions. Littlestone dimension controls ordinary online learnability, but proper repair dimension controls actionable online buying. A concept class can be easy to learn in the standard mistake-bound sense and still be hard to maintain as a live actionable prediction once the algorithm is required to stay proper and to buy the realized action online. For a finite binary concept class, this paper defines the proper repair dimension PRD(H) by a dynamic program on version spaces. The value PRD(H) is exactly the optimal deterministic worst-case number of repairs for a proper learner that must keep a live hypothesis through every realizable labeled sequence. The paper proves Ldim(H) <= PRD(H) <= |H|-1, with tight examples: the full class on d coordinates has Ldim = PRD = d, while the universal coordinate class U_n has Ldim(U_n) = floor(log_2 n) and PRD(U_n) = n-1. This gap transfers directly to online buying. The paper builds a unit-cost actionable buying instance from every proper class H and proves that the terminal benchmark is uOPT = 1 while every deterministic proper actionable algorithm pays exactly 1 + PRD(H) in the worst case. For U_{2^d}, this gives deterministic cost 2^d despite Littlestone dimension d. The paper also gives a positive transfer theorem for componentized actionable prediction classes, showing that bounded repair dimension together with bounded PRD-load congestion yields controlled actual buying cost.
