Learning curves are a fundamental primitive in supervised learning, describing how an algorithm's performance improves with more data and providing a quantitative measure of its generalization ability. Formally, a learning curve plots the decay of an algorithm's error for a fixed underlying distribution as a function of the number of training samples. Prior work on revenue-maximizing learning algorithms, starting with the seminal work of Cole and Roughgarden [STOC, 2014], adopts a distribution-free perspective, which parallels the PAC learning framework in learning theory. This approach evaluates performance against the hardest possible sequence of valuation distributions, one for each sample size, effectively defining the upper envelope of learning curves over all possible distributions, thus leading to error bounds that do not capture the shape of the learning curves. In this work we initiate the study of learning curves for revenue maximization and provide a near-complete characterization of their rate of decay in the basic setting of a single item and a single buyer. In the absence of any restriction on the valuation distribution, we show that there exists a Bayes-consistent algorithm, meaning that its learning curve converges to zero for any arbitrary valuation distribution as the number of samples $n \to \infty$. However, this convergence must be arbitrarily slow, even if the optimal revenue is finite. In contrast, if the optimal revenue is achieved by a finite price, then the optimal rate of decay is roughly $1/\sqrt{n}$. Finally, for distributions supported on discrete sets of values, we show that learning curves decay almost exponentially fast, a rate unattainable under the PAC framework.

翻译：学习曲线是监督学习中的基本概念，它描述了算法性能如何随数据增加而提升，并为其泛化能力提供了定量度量。形式上，学习曲线绘制了算法针对固定潜在分布的误差衰减情况，该误差是训练样本数量的函数。关于收益最大化学习算法的先前研究始于Cole和Roughgarden的经典工作（STOC, 2014），采用了独立于分布的视角，这与学习理论中的PAC学习框架相呼应。该方法评估算法在面对所有可能分布中最困难的估值序列时的性能（每个样本大小对应一个分布），实际上定义了所有可能分布下学习曲线的上包络线，从而导致无法捕捉学习曲线形状的误差界。本文首次对收益最大化的学习曲线进行研究，并在单一物品和单一买家的基本设置中近乎完整地表征了其衰减速率。在没有对估值分布施加任何限制的情况下，我们证明存在一个贝叶斯一致算法，即对于任意估值分布，当样本数量$n \to \infty$时，其学习曲线收敛到零。然而，即便最优收益是有限的，这种收敛也必然任意缓慢。相比之下，如果最优收益由有限价格实现，则最优衰减速率约为$1/\sqrt{n}$。最后，对于支撑集为离散值集合的分布，我们证明学习曲线几乎以指数级速度衰减，这一速率在PAC框架下无法实现。