We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the Exponential Time Hypothesis fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. Moreover, we show that this MIP formulation is ideal (i.e. the strongest possible formulation) for the revenue function of a single impression. Since it can be computationally expensive to exactly solve the MIP formulation in practice, we also study the performance of its linear programming (LP) relaxation. Though it may work well in practice, we show that, unfortunately, in the worst case the optimal objective of the LP relaxation can be O(number of samples) times larger than the optimal objective of the true problem. Finally, we present computational results, showcasing that the MIP formulation, along with its LP relaxation, are able to achieve superior in-and out-of-sample performance, as compared to state-of-the-art algorithms on both real and synthetic datasets. More broadly, we believe this work offers an indication of the strength of optimization methodologies like MIP to exactly model intrinsic discontinuities in machine learning problems.

Summary and Contributions: In the second-price auction with reserve price, the publisher sets a reserve price before the auction, and the payment of the winner is defined as the maximum of the second highest bid and the reserve price. The paper deals with the problem of deciding the reserve price from the side information to maximize the payment from the winner. It considers using linear regression to infer the best reserve price. Then the problem is reduced to the optimization problem of optimizing the model parameters so as to maximize the payment. The main contribution of this paper is to discuss how to solve this optimization problem.

We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the Exponential Time Hypothesis fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. Moreover, we show that this MIP formulation is ideal (i.e. the strongest possible formulation) for the revenue function of a single impression. Since it can be computationally expensive to exactly solve the MIP formulation in practice, we also study the performance of its linear programming (LP) relaxation. Though it may work well in practice, we show that, unfortunately, in the worst case the optimal objective of the LP relaxation can be O(number of samples) times larger than the optimal objective of the true problem.