On the Learning Curves of Revenue Maximization

Problem
This paper addresses a gap in the understanding of learning curves specifically for revenue-maximizing algorithms in supervised learning. Prior research has primarily focused on distribution-free perspectives that do not adequately characterize the shape of learning curves, particularly in the context of revenue maximization. The authors initiate a study of learning curves for revenue maximization, providing a near-complete characterization of their decay rates in a simplified setting involving a single item and a single buyer. This work is presented as a preprint and has not yet undergone peer review.

Method
The authors propose a Bayes-consistent algorithm that demonstrates convergence of the learning curve to zero for any arbitrary valuation distribution as the number of samples ( n ) approaches infinity. They analyze the decay rates of learning curves under different conditions: when the optimal revenue is finite and achieved at a finite price, the decay rate is approximately ( 1/\sqrt{n} ). In cases where the valuation distribution is supported on discrete sets of values, the authors show that the learning curves decay almost exponentially fast. The paper does not disclose specific architectural details or training compute requirements, focusing instead on theoretical characterizations.

Results
The paper presents several key findings regarding the decay rates of learning curves for revenue maximization. The main results include:

• A Bayes-consistent algorithm achieves convergence to zero for arbitrary valuation distributions, albeit at an arbitrarily slow rate.
• When optimal revenue is finite and achieved at a finite price, the decay rate of the learning curve is ( O(1/\sqrt{n}) ).
• For discrete valuation distributions, the learning curves exhibit an almost exponential decay rate, which is significantly faster than the rates achievable under the PAC learning framework. These results provide a quantitative measure of performance improvement with increasing data, contrasting with previous distribution-free approaches.

Limitations
The authors acknowledge that their findings are limited to the basic setting of a single item and a single buyer, which may not generalize to more complex scenarios involving multiple items or buyers. Additionally, the convergence rates, while theoretically significant, may not be practically achievable in real-world applications where valuation distributions can be complex and dynamic. The paper does not explore the implications of these limitations on the applicability of the proposed algorithm in broader contexts.

Why it matters
This work has significant implications for the design and analysis of revenue-maximizing algorithms in machine learning. By providing a detailed characterization of learning curves, it enhances the understanding of how performance scales with data in revenue contexts, which is crucial for applications in auction design, pricing strategies, and economic modeling. The results challenge existing paradigms in learning theory and suggest new avenues for research in revenue maximization, particularly in exploring more complex buyer and item interactions.

Authors: Steve Hanneke, Alkis Kalavasis, Shay Moran, Grigoris Velegkas
Source: arXiv:2604.26922
URL: https://arxiv.org/abs/2604.26922v1