Max Biggs

Max Biggs, Georgia Perakis

We focus on modeling the relationship between an input feature vector and the predicted outcome of a trained decision tree using mixed-integer optimization. This can be used in many practical applications where a decision tree or tree ensemble is incorporated into an optimization problem to model the predicted outcomes of a decision. We propose tighter mixed-integer optimization formulations than those previously introduced. Existing formulations can be shown to have linear relaxations that have fractional extreme points, even for the simple case of modeling a single decision tree. A formulation we propose, based on a projected union of polyhedra approach, is ideal for a single decision tree. While the formulation is generally not ideal for tree ensembles or if additional constraints are added, it generally has fewer extreme points, leading to a faster time to solve, particularly if the formulation has relatively few trees. However, previous work has shown that formulations based on a binary representation of the feature vector perform well computationally and hence are attractive for use in practical applications. We present multiple approaches to tighten existing formulations with binary vectors, and show that fractional extreme points are removed when there are multiple splits on the same feature. At an extreme, we prove that this results in ideal formulations for tree ensembles modeling a one-dimensional feature vector. Building on this result, we also show via numerical simulations that these additional constraints result in significantly tighter linear relaxations when the feature vector is low dimensional. We also present instances where the time to solve to optimality is significantly improved using these formulations.

Max Biggs

We study an off-policy contextual pricing problem where the seller has access to samples of prices that customers were previously offered, whether they purchased at that price, and auxiliary features describing the customer and/or item being sold. This is in contrast to the well-studied setting in which samples of the customer's valuation (willingness to pay) are observed. In our setting, the observed data is influenced by the previous pricing policy, and we do not know how customers would have responded to alternative prices. We introduce suitable loss functions for this setting that can be directly optimized to find an effective pricing policy with expected revenue guarantees, without the need for estimation of an intermediate demand function. We focus on convex loss functions. This is particularly relevant when linear pricing policies are desired for interpretability reasons, resulting in a tractable convex revenue optimization problem. We propose generalized hinge and quantile pricing loss functions that price at a multiplicative factor of the conditional expected valuation or a particular quantile of the prices that sold, despite the valuation data not being observed. We prove expected revenue bounds for these pricing policies respectively when the valuation distribution is log-concave, and we provide generalization bounds for the finite sample case. Finally, we conduct simulations on both synthetic and real-world data to demonstrate that this approach is competitive with, and in some settings outperforms, state-of-the-art methods in contextual pricing.

Ruijiang Gao, Max Biggs, Wei Sun et al.

Unlike traditional supervised learning, in many settings only partial feedback is available. We may only observe outcomes for the chosen actions, but not the counterfactual outcomes associated with other alternatives. Such settings encompass a wide variety of applications including pricing, online marketing and precision medicine. A key challenge is that observational data are influenced by historical policies deployed in the system, yielding a biased data distribution. We approach this task as a domain adaptation problem and propose a self-training algorithm which imputes outcomes with categorical values for finite unseen actions in the observational data to simulate a randomized trial through pseudolabeling, which we refer to as Counterfactual Self-Training (CST). CST iteratively imputes pseudolabels and retrains the model. In addition, we show input consistency loss can further improve CST performance which is shown in recent theoretical analysis of pseudolabeling. We demonstrate the effectiveness of the proposed algorithms on both synthetic and real datasets.

Max Biggs, Ruijiang Gao, Wei Sun

We study a pricing setting where each customer is offered a contextualized price based on customer and/or product features. Often only historical sales data are available, so we observe whether a customer purchased a product at the price prescribed rather than the customer's true valuation. Such observational data are influenced by historical pricing policies, which introduce difficulties in evaluating the effectiveness of future policies. The goal of this paper is to formulate loss functions that can be used for evaluating pricing policies directly from observational data, rather than going through an intermediate demand estimation stage, which may suffer from bias. To achieve this, we adapt ideas from machine learning with corrupted labels, where we consider each observed purchase decision as a known probabilistic transformation of the customer's valuation. From this transformation, we derive a class of unbiased loss functions. Within this class, we identify minimum variance estimators and estimators robust to poor demand estimation. Furthermore, we show that for contextual pricing, estimators popular in the off-policy evaluation literature fall within this class of loss functions. We offer managerial insights into scenarios under which these estimators are effective.

Max Biggs, Wei Sun, Markus Ettl

Data-driven pricing strategies are becoming increasingly common, where customers are offered a personalized price based on features that are predictive of their valuation of a product. It is desirable for this pricing policy to be simple and interpretable, so it can be verified, checked for fairness, and easily implemented. However, efforts to incorporate machine learning into a pricing framework often lead to complex pricing policies which are not interpretable, resulting in slow adoption in practice. We present a customized, prescriptive tree-based algorithm that distills knowledge from a complex black-box machine learning algorithm, segments customers with similar valuations and prescribes prices in such a way that maximizes revenue while maintaining interpretability. We quantify the regret of a resulting policy and demonstrate its efficacy in applications with both synthetic and real-world datasets.