Best of Many in Both Worlds: Online Resource Allocation with Predictions under Unknown Arrival Model

TL;DR

The extent to which any algorithm can optimally leverage predictions is tightly characterized, via a formal lower bound, the extent to which any algorithm can optimally leverage predictions without knowing the prediction accuracy or the underlying arrival model.

Abstract

Online decision-makers often obtain predictions on future variables, such as arrivals, demands, inventories, and so on. These predictions can be generated from simple forecasting algorithms for univariate time-series, all the way to state-of-the-art machine learning models that leverage multiple time-series and additional feature information. However, the prediction accuracy is unknown to decision-makers a priori, hence blindly following the predictions can be harmful. In this paper, we address this problem by developing algorithms that utilize predictions in a manner that is robust to the unknown prediction accuracy. We consider the Online Resource Allocation Problem, a generic model for online decision-making, in which a limited amount of resources may be used to satisfy a sequence of arriving requests. Prior work has characterized the best achievable performances when the arrivals are either generated stochastically (i.i.d.) or completely adversarially, and shown that algorithms exist which match these bounds under both arrival models, without ``knowing'' the underlying model. To this backdrop, we introduce predictions in the form of shadow prices on each type of resource. Prediction accuracy is naturally defined to be the distance between the predictions and the actual shadow prices. We tightly characterize, via a formal lower bound, the extent to which any algorithm can optimally leverage predictions (that is, to ``follow'' the predictions when accurate, and ``ignore'' them when inaccurate) without knowing the prediction accuracy or the underlying arrival model. Our main contribution is then an algorithm which achieves this lower bound. Finally, we empirically validate our algorithm with a large-scale experiment on real data from the retailer H&M.

Figures (3)

• Figure 1: (Figure and caption from an2023nonstationary) Daily number of customers (in blue), from September 2014 to January 2015, at two different stores in the Rossmann drug store chain. Predictions (in red), starting November 2014, are generated using Exponential Smoothing with the same fitting process. The store in the upper sub-figure has substantially more accurate predictions ($R^2=0.88$) than that of the lower sub-figure ($R^2=0.11$).
• Figure 2: Two potential arrival sequences for an online resource allocation problem with a single resource (two lemons) and two time periods. The left (right) sequence falls under the stochastic (adversarial) arrival model.
• Figure 3: Histograms of GAPs with different forecasting methods, each containing 100 instances.

Theorems & Definitions (18)

• Proposition 6: Lower Bound, Informal
• Theorem 1: Upper Bound, Informal
• Proposition 1: "Perfect" Dual Variable
• Definition 1: Measure of Stationarity
• Proposition 3: Theorem 1 and Theorem 2 in balseiro2023best
• Proposition 4: Corollary of Theorem 2 in orabona2013dimension
• Proposition 5: Corollary of Theorem 3.1 in balseiro2019learning
• Theorem 1: Upper Bound
• Proposition 6: Lower Bound
• Proposition 7